Categories
Themes in Mathematics

What is the Fibonacci series, and to what extent is it true?

Italian Mathematician Leonardo Pisano( born in 1175 and died around 1250) also known as Fibonacci is mostly famous for his Fibonacci sequence. His name actually got originated from a misreading on a manuscript of “filius Bonacci”(son of Bonaccio).

Now discussing on his exceptional work on Fibonacci Sequence.The name “Fibonacci sequence” was first applied by Theorist Edouard Lucas in the 19th century.

In the field of Mathematics, Fibonacci numbers denoted as F_n.The sequence states that that each number is the sum of the two preceding numbers starting from 0 followed by 1.

The general term of the sequence

\(F_n\)=\(F_{n−1}\)+\(F_{n−2}\) where \(F_0\)=

(0\)and \(F_1\)=\(1\) \(∀\) \(n>1\)

Thus the sequence becomes

0,1,1,2,3,5,8,13,21,34, and so on.

Fibonacci discovered a very interesting concept of the rabbit population.Rabbits usually never die and they are able to reproduce at the end of its second month.

Now if a male and a female rabbit that is a newly born pair of rabbits are placed in a field then they will always produce a new pair at the end of each month starting from the second month.

This way the following observations were made.

  1. By the end of first month there is only one pair. \(F_1\)=\(1\))
  2. Now By the end of second\d month, a new pair is born thus amounting to 2 pairs \(F_2\)=\(2\))
  3. By the end of third month a new pair is born from the original pair thus amounting to 3 pairs \(F_3\)=\(F_2\)+\(F_1\)=\(2+1=3\))
  4. By the end of fourth month again a new pair is born from the original pair and another pair is born from the first female produced by the original female amounting to 5 pairs \(F_4\)=\(F_3\)+\(F_2\)=\(3+2=5\))

We can conclude from the above mentioned facts that by the end of n month, the number of pairs will be

\(F_n\)=\(F_{n−1}\)+\(F_{n−2}\), which is the Mathematical generalised expression of the Fibonacci Sequence.

This fact about rabits is a big example of it being true. It is true in the sense it does exist in nature.

Now lets discuss the relation of Fibonacci Sequence and the Golden Ration and few applications of it in nature

Two numbers are said to be in Golden ratio if they are in the ratio of the sum of the numbers to the larger number. Most of the things in nature occurs according to this ratio including the spiral arrangement of leaves . The value of Golden ratio 1.618.1.618.Fibonacci Sequence and Golden ratio are interlinked

the Fibonacci sequence is 0,1,1,2,3,5,8,13,21,34,………55………

Now going by the golden ratio dividing each number by previous number we get \(\frac{1}{1}\)=\(1\) , \(\frac{2}{1}\)=\(2\) , \(\frac{3}{2}\)=\(1.5\) , \(\frac{5}{3}\)= \(1.6666\), \(\frac{8}{5}\)= \(1.6\), \(\frac{13}{8}\) = \(1.625\) , \(\frac{21}{13}\)= \(1.61\) , \(\frac{34}{21}\)=\(1.619\) , \(\frac{55}{34}\)=\(1.617\). We notice the values are converging towards 1.618 which is the golden ratio.

Most of the things in nature follows the Golden Ratio and the Fibonacci sequence.

The spiraling pattern of the sunflower and Pine cones follows the Fibonacci sequence.The branches of tree are also arranged in Fibonacci sequence. Even the galaxies follows Fibonacci pattern.Another very interesting fact about the Fibonacci number is that number of petals on flower Daisy is always a Fibonacci number (21,34,55) being most common numbers).The number of petals of flowers are often arranged in Fibonacci numbers. Like lilies have 3, Chicory have 21,daisies have 13 and all these are Fibonacci numbers.

Physics. In optics the number of different beam paths when a ray of light shines at an angle through two different transparent plates of different refractive index and material, there are k reflections ,for \(k>1\) and k is the Fibonacci number.

As recorded,1597,was the last year that was a Fibonacci number and the next will be 2584.

Categories
Themes in Mathematics

How was the mathematical number system formed? How was this “odd” & “even ” concept started in mathematics?

The number system is the numeral representation of numbers in It is formed by using digits or various Mathematical symbols.Now there are three types of numeral systems.

  • Decimal Numeral/Number System- It is the system used for notifying integers or non integers.This system is also referred to as the base ten positional numeral system.Decimals are mainly identified having a decimal separator which is a ‘.’ . For example 3.14 ,7.00 etc.The number representation in the decimal system are in the form of decimal fractions (p/(10^{q}) ,where p is an integer and q is a non negative integer.

Decimal Numbers are counted this way

000 , 001 , 002 , 003 , 004 , 005 , 006, 007 , 008 ,009 , 010………….100………1000 and so on .

  • Binary Number System – This is a base 2 number system usually used in Mathematics as well as Computers. It comprises only two symbols 0 and 1 . It has very direct flow of representations and is widely used in logic gates in digital electronics. Binary number counting only includes zeros and ones . For example binary number for 2 is 10. On diving 10 by 2 we get the remainder 0 and the quotient 5 and on dividing 5 by 2 again we get the remainder 1 and quotient 2. Now rearranging the remainders in reverse order we get 10 .
  • Unary Numeral System – This is a base one numeral system.This is the simplest way for representing natural numbers. The way of representation of N is representation of a symbol exactly N number of times. For example the unary system for 5 is 11111 and 4 is 1111 if 1 is the chosen symbol for the representation.

The most common representation of numbers is the Hindu Arabic Numeral System developed by Aryabhata and Bramhagupta. In Mathematics there is a term called the Decimal place Value System also called Positional Notation. This concept of place value notation was developed by Aryabhata and is the backbone of the number system because without place value notation we cannot represent numbers and Bramhagupta first used the symbol for Zero .Again without Zero number theory is not possible. These two concepts although developed India later was widely spread-ed in other countries and the Arabs modified it.

As mentioned above the simplest numeral system is the Unary numeral system used for tallying marks and scores and coding purposes.Lets use the unary representation for larger numbers. For example 543 if – stands for units / for tens and + for hundreds then the representation will be +++++////— for 543. This system is also referred to as the sign modification System and the Roman Numeral System is a modification of this concept.

Place value System or Positional Notation System is a very useful and demanding concept of Number System.This means that the value of a number is determined by the position of the digit that is the value of a number is actually the product of the digit by a factor which is determined by the position of the digit.For example lets take three identical digits 999. Here the interesting part is in words the number is written as nine hundred and ninety nine . The hundreds tens and the units here are being determined by the position of the digits that is digit at the first place represents the units, second place represents the tens and the third place represents hundreds. Similarly any digit at the fourth place shall reprimand thousands.

In computers the mainly used number systems are the binary number system(base 2) , Octal Number System(grouping of binary digits by 3) and hexadecimal number system( grouping of binary digits by 4 ).

The Hindu texts on numerals were translated by the Arabs and was spread throughout the Western World. Further modifications were done by the Western Culture and were called the Arabic Numerals. This widely helped in spreading the Mathematical Number System throughout the World.

Now this idea of dividing numbers into two kinds odd(not divisible by 2) , even (divisible by 2) is called Parity. The generalized way of representing odd numbers is 2k+1 , where k is an integer and even numbers is 2k , where k is an integer. Some examples of even numbers are 2, 4, 6, 8 ,54 ,78 ,-2, -18 and some examples of odd numbers are 3 ,9 ,17, 43, 19, -5.

Some basic rules for odd and even numbers include –

  • Addition – Addition of two even numbers is even and odd numbers is even. for example 2+4=6 (both 2 and 4 are even numbers and the resultant 6 is also an even number) . Similarly 13+15 = 28 ( both 13 and 15 are odd numbers and the result is an even number) where as addition of an even and odd number is odd. Example 17+2= 19 (17 is an odd number and 2 is an even number but the result is odd)
  • Subtraction- The rules for subtraction is same as in addition.
  • Multiplication- Product of two even numbers is even, two odd numbers is odd and product of an even and odd number is even.
  • Division- In this case if the quotient is an integer and the and the dividend has more factors of two than the divisor then only the result will be even.

0 is considered to be an even number.

Disclaimer : The history of odd and even numbers that is their origin is not properly known to me

Categories
Themes in Mathematics

Who needs mathematics?

Well! That is a pretty interesting question nut the question has only one answer. Mathematics is not a need but a way of life. What is the definition of Mathematics? The most simplest answer would be a subject including numbers and number related topics but is it true? Is Mathematics limited to just a subject? No. It defines life .Where do we not find Mathematics?

Let us take few interesting examples-

We all love cricket right? Its my favorite sport too. Now how many players is the Indian Cricket Team comprised of? 11 and how do we count it ,Mathematics.Counting is one of the best and most important application of Mathematics because without the knowledge of numbers how is counting possible?

Now shopping, we all do marketing and shopping in a daily basis. So while paying bills we or asking for changes from the shopkeeper, Mathematics is the basic application.There comes the role of the prime topics in Mathematics which are Addition subtraction Division Multiplication.

Travelling to a certain place within a scheduled time. What should be our speed to reach the destination within time is again Mathematics AND THE TOPIC IS SPEED,DISTANCE ,TIME.

The revolution and rotation of Earth.Its time period that is Earth rotates in its axis in about 24 hours that makes the conclusion 1day = 24 hours.

Earth revolves around the Sun in 365 days, and the conclusion is 1 year= 365 days.

Calculating the area of a land in square meters or any other unit, how much water a drum can contain, how much boundary is needed to cover a particular place,capacity of a tank,everything is Mathematics. Mathematics is not a subjects it is a routine that follows us every.Life is incomplete without this topic.

Indirectly each and every human being on Earth is using Mathematics.

The medicines we use includes Mathematics too as which medicine will contain how much gram of which liquid or any other medicinal item.Today the mobile phones and all other electgronic appliances we are using has Mathematics everywhere.Computing,Cryptography ,Encryption,Programming theses things work in our Mobile phones,ATMs, Computers,Laptops and without Mathematics these operations cannot be performed.

Even the cars ,trucks,buses,cycles, Airplanes, Trains,Jets,Rockets all these includes Mathematics in their mechanism. The correct radius and diameter of the tires, petrol and diesel content,weight of the vehicles all these thing is it possible without Mathematics?

Therefore Mathematics has no limited users. Mathematics is not a need but it is like a source of life.

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AMC 10 USA Math Olympiad

AMC 10A Year 2014 Problem 20 Sequential Hints

Understand the problem

The product \\((8)(88888……8)\\), where the second factor has k digits, is an integer whose digits have a sum of \\(1000\\). What is k? $\\textbf{(A)}\\ 901\\qquad\\textbf{(B)}\\ 911\\qquad\\textbf{(C)}\\ 919\\qquad\\textbf{(D)}\\ 991\\qquad\\textbf{(E)}\\ 999$

Source of the problem
American Mathematical Contest 10A Year 2014

Topic

Number Theory 

Difficulty Level

7/10

Suggested Book

Problem Solving Strategies  Excursion In Mathematics 

Start with hints

Do you really need a hint? Try it first!

After having a long look into this problem you can first make attempt by listing the first few numbers of the given form.Give it a try!!!!!

So we can do it like this  8*(8)=64 8*(88)=704 8*(888)=7104 8*(8888)=71104 8*(88888)=711104 Now try to observe the pattern in the above table because here lies the main insight of this problem . Come on cook it up!!!!!!    

So form the table you can observe the terms are following a pattern that’s is The first number is 7 Then k-2 number of 1 Then the last two digits are 04 

Now try to make the sum to 1000

So now you are in the final part so you can easily find  7+04+(k-2)=1000

implies 11+(k-2)=1000 . Solving this equation we get the value of K is 991 which is the required answer.

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Themes in Mathematics

What are some mind-blowing facts about mathematics?

Mathematics is itself mind blowing. It is not only a subject or a topic but it is the description of life. Nothing is possible without Mathematics. Mathematics is not only about interest but it is about excellence. People often misguide the subject as “full of problems” but without problems is there any solution?

Lets begin with the definition of Mathematics. It can be stated as a subject including topics on algebra, geometry,number theory,arithmetic,analysis,geometry and many more. But in reality there is no definition of Mathematics. This is such a path which can be traveled only if you start visualizing its properties. It is all about measurement,counting,conjectures,analyzing,numbers,calculation and most importantly logic.

Now coming to the question some of the mind blowing facts about Mathematics

  • Invention of ZERO, one of the most important discoveries till date is zero and zero is actually an even number.
  • Hundred is derived from the word “Hundrath” which in Old Norse meant 120 meaning Long Hundred.
  • There are a number of prime numbers but 2 is the only ‘even’ prime number.
  • There is no Roman numeral representation of Zero where as all oher numbers can be written in Roman Numerals.
  • 4 is the only number spelled with the numbers of letters equal to its value.
  • Aryabhatta did not provide any symbol to Zero. The symbol of Zero was used in an Arithmetic manual of merchants, Bakhshali Manuscript.
  • The Fibonacci sequence invented by Leonardo Fibonacci has practical applications in nature. Breeding of Rabbit follows the Fibonacci sequence. The sequence is given as 0,1,1,2,3,5….. where each term is the sum of preceding terms.The spiraling pattern of the sunflower and Pinecones follows the Fibonacci sequence.The branches of tree are also arranged in Fibonacci sequence. Even the galaxies follows Fibonacci pattern.
  • 10! seconds = 6 weeks.

Lets check the above fact if its true or not

Value of 10! = \(1*2*3*4*5*6*7*8*9*10\) = 3,628,800 seconds

Now converting to minutes we get \(\frac{3628800}{60}\)= 60,480 minutes( 1 minute = 60 seconds)

Converting to hours \(\frac{60480}{60}\) = 1,008 hours (1 hour= 60 minutes)

Converting to days \(\frac{1008}{24}\) = 42 days (1 day= 24 hours)

6 weeks =\( 6*7\)= 42 days ( 1 week = 7 days)

Now the number 11711 is a Palindrome number, that is numbers that read the same from forwards and backwards.

  • First Electronic Calculator was created in 1960s.
  • A 100-sided polygon is called a hectogon OR centagon.
  • Two numbers are said to be in Golden ratio if they are in the ratio of the sum of the numbers to the larger number. Most of the things in nature occurs according to this ratio including the spiral arrangement of leaves . The value of Golden ratio 1.618.
  • Fibonacci Sequence and Golden ratio are interlinked

  • The Fibonacci sequence is 0,1,1,2,3,5,8,13,21,34,55………

Now going by the golden ratio dividing each number by previous number we get \(\frac{1}{1}\)=1 , \(\frac{2}{1}\)=2 , \(\frac{3}{2}\)=1.5 , \(\frac{5}{3}\)= 1.6666, \(\frac{8}{5}\)= 1.6, \(\frac{13}{8}\) = 1.625 , \(\frac{21}{13}\)= 1.61 , \(\frac{34}{21}\)=1.619 , \(\frac{55}{34}\)=1.617

We notice the values are converging towards 1.618 which is the golden ratio.

  • The number 5040 has unique properties that is , this number is the sum of 42 primes that is 42 consecutive primes and that is 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 +163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229.

  • The number 3 has various properties like

It is the first prime number which is odd.

It is a factorial prime number .A factorial prime is a prime number which is one less pr more than a factorial

3 is a factorial prime because 3!= (2!+1) = 6

It is a Mersenne prime number too.Now in mathematics Mersenne prime number is a prime number which is 1 less than a power of 2 that is

\(2^n\) – 1 , where n is an integer.

3 can be written as 3= \(2^2\) – 1 = 4-1

Now MATHEMATICS can be divided into two parts-

  1. Pure Mathematics
  2. Applied Mathematics

Pure Mathematics is devoid of any applications outside Mathematics. Its solely pure containing deeper aspects and a vivid study on Mathematics.The results obtained from pure Maths are greatly useful for practical applications.Concept of pure Mathematics was obtained around the year 1900.Most of the Mathematical theories are obtained from real world and less from the applied world.

Topics under Pure Mathematics includes

  1. Algebra( factorization, Group theory, Ring Fields, Metric space,Gaussian space, Theory of Equations,Binary Operations)
  2. Calculus( limits,continuity, differentiation, integration, relations and functions)
  3. One of the best of Mathematics Number Theory( Fermat’s Theorem, Mean value Theorem, Euler’s theorem phi function, Lagrange’s Theorem, Integers, Real Numbers, Complex Numbers, Rational and Irrational Numbers, Quadratic Equations, Cardan;s Method, Ferrari’s Method, Mathematical Induction, Polynomials,Topology,Real and Complex Analysis).
  4. Logic( set theory,cardinal,Boolean Algebra)
  5. Combinatorics
  6. Geometry (Planes,Straight Lines,Curves, Polynomials,Construction, angles)

Applied Mathematics is the application of Mathematical topics in various fields like Engineering, Business, Industry, Computer Science. The term “Applied” means applying Mathematics in professional fields by studying Mathematical models.

Topics under Applied Mathematics include

  1. Differential Equations , a Mathematical Equation containing unknown functions and derivatives. It is mostly comprised of Calculus. There are two types of differential equations- Ordinary Differential Equation (contains one or more functions of one independent variable and derivative of those functions) and Partial Differential Equations (involves derivatives in multiple variables).
  2. Mathematical Physics ( Quantum Theory, Wave Theory, Ray optics etc)
  3. Statistics and Probability.

Well, going into a deeper thought not all of the things around us revolves around Mathematics?

Lets consider our daily routine , Marketing and shopping without numbers or basic Arithmetic is shopping possible?No then what are numbers and money all about? Mathematics. Reaching a place within time let it be school or any other work place, what is time about? Hours seconds and minutes!! Again Mathematics. Speed Distance and time dont we apply it in daily life? This is also Mathematics. Profit and Loss, Sales Tax, Value Added Tax everything is about Mathematics. Thus Mathematics is not just a subject but it is a visualisation of our daily life.

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Themes in Mathematics

Who was the best mathematician in the initial era?

Alright! So there are a number of Mathematicians who have enriched the field of Mathematics like Pythagoras,Leonardo Pisano, Srinivasa Ramanujan, Euler.

Not really pointing at the best but one of the best Mathematicians till date we have is Johann Carl Friedrich Gauss. Johann Carl Friedrich Gauss, born on 30th April 1777 and died on 23rd February 1855 at the age of 77 was a German Mathematician, He was a brilliant Mathematician and is considered to be the most influential Mathematicians in History.He is often regarded as the “Prince of Mathematics” and “Foremost of Mathematicians”.

Gauss was born on Brunswick and he belonged to a poor family. His supreme intelligence captured the attention of the Duke of Brunswick who sent him to the Collegium Carolinum (now, known as the Braunschweig University of Technology and to University of Gottingen. He attended the Collegium Carnolium from the year 1792-1795 and Gottingen from 1795-1798. Gauss had child prodigy, means a person under ten age of ten having exceptional knowledge and outputs on a particular domain like that of an adult. In 1796, Gauss came up with the discovery that if the number of sides of a polygon occurs as the product of distinct Fermat primes and a power of 2, then a regular polygon can be constructed using a compass and straightedge. Fermat number is referred to as a positive integer of the form

\(F_n\) = \(2^{2n}\)+1, where n is an integer. Some Known Fermat Numbers are 3,5,17,257,……. The Fermat numbers which are primes are known as Fermat primes.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root (includes polynomial having real coefficients).

In his book ‘Disquisitiones Arithmeticae ‘,meaning Arithmetical Investigations in Latin,Gauss worked a lot on number theory and found the symbol of congruence used in modular arithmetic.

Gauss proved a number of theorems like

  1. Fermat’s polygonal number theorem for n=3
  2. Descarte’s rule of signs.
  3. Algorithm for determining date of ester
  4. Fermat’s last theorem for n=5.

Fermat’s polygonal number theorem for n=3

This theorem states that every positive integer can be expressed as the sum of three or more triangular numbers, three or more square numbers or three or more pentagonal numbers.

Triangular number-p is called a triangular number if it is the number of dots that can be arranged in a triangle having p dots on a side and equals to the sum of p natural numbers from 1 to p. example- 31,3,6,10,15,21……(here p is any natural number)

Square numbers refers to those numbers which are a perfect square of an integer. Example ‘9’, as it is a square of 3*3.

Pentagonal numbers are just an extension of square and triangular numbers but the arrangement of dots is different in this case. Let p_n be nth pentagonal number where n is the number of dots \(n>=1\) Then \(p_n\) is given by the formula

\(p_n\) = \(\frac{3(n^2)-n}{2}\)

The triangular concept was proved by Gauss in 1796. Gauss’s results are often referred to as the Eureka Theorem.

Descarte’s Rule of Signs

This is a method to find the number of positive real roots of a polynomial. It states that the number of positive roots of a polynomial is the number of sign changes in the polynomial sequence excluding the zero coefficients and the difference between the numbers is always even.

Example

\(f(x)\) = + \((x^3)\) + \((x^2)\) – \((x)\) -2

Here sign change is occurring jut one time between the second and third term. Therefore this polynomial has one positive root.

Fermat’s last theorem for n=5

Fermat’s last theorem by Pierre de Fermat states that no three integers can satisfy the equation

\(a^n\)+ \(b^n\)=\(c^n\) , where a,b,c, are positive integers and n is any integer greater than 2

This above theorems were proved by Gauss.

Now there were several Mathematicians of the initial era as mentioned above but among them according to me Gauss was the best because of his ravishing outlook towards Mathematics. The famous Easter calculation concept was also provided by him. Easter is a festival which was celebrated first three days after Crucifixion of Jesus Christ. Initially it was believed that there were no fixed dates for Easter in the calendar every year. Gauss obtained this calculation using modular Arithmetic. Thus his way of connecting each and every saga of life with Mathematics is extremely unique.

Some other famous Mathematicians are

  1. Thales of Miletus (considered as the first true Mathematician).
  2. Pythagoras of Samos, creator of famous Pythagoras theorem.
  3. Leonardo Pisano, inventor of Fibonacci series.
  4. Euclid, famously mentioned as the Father of Geometry.
  5. Leonhard Euler, one of the greatest Mathematicians ever existed.
  6. Srinivasa Ramanujan, had huge contributions towards Mathematical Analysis, Continued Fractions, Number Theory.
  7. Pierre de Fermat, best known for Fermat’s principle of light and Fermat’s last theorem.
  8. Christian Goldbach, famous for his still unsolved Goldbach conjecture.
  9. Georg Ferdinand Ludwig Philipp Cantor , creator of set theory.
  10. Sir Issac Newton, developer of infinitesimal calculus.
  11. Jules Henri Poincaré ,described as the “The Last Universalist,”
  12. Georg Friedrich Bernhard Riemann, one of the greatest contributors of differential geometry.
  13. Augustin-Louis Cauchy,had great contributions in the line of Mathematical analysis.
  14. Diophantus of Alexandria,famous for his study on Diophantine equations.
  15. Pierre-Simon, marquis de Laplace ,formulator of Laplace equation and Laplace transformation.
  16. Lodovico Ferrari. solver of quarter equations.
  17. Gerolamo Cardano,well known in the field of Algebra and solver of cubic equations.

One of Gauss’s popular quotes

“It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.”

which i truly believe.

Categories
AMC 10 USA Math Olympiad

AMC 10A Year 2005 Problem 21 Sequential Hints

Understand the problem

For how many positive integers \(n\) does 1+2+3+4+….+n evenly divide from 6n?

(a)3.       (b)5.       (c)7.       (d)9.       (e)11

Source of the problem
American Mathematical Contest 2005 10A Problem 21

Topic
Number Theory 

Difficulty Level
6/10

Suggested Book
Challenges and Thrills in Pre College Mathematics

Excursion Of Mathematics 

Do you really need a hint? Try it first!

Step 1.

So after having a deep look into this problem you can see that if 1+2+3+…..+n evenly divides 6n that is \(\frac{6n}{1+2+3+….+n}\) now to think about formula of the sum of 1+2+3+…..+n.

 

 

Step 2.

After getting the formula as 1+2+3+4+….+n=\(\frac{n(n+1)}{2}\) substitute it in the equation \(\frac{6n}{1+2+3+….+n}\) and simplify it. Give it a try!!!!!!

Step 3

Now by simplifying you will get \(\frac{12}{n+1}\). Now here lies the main concept of this problem as you have to find integer n so you must see that if (n+1) is a factor of 12 then only \(\frac{12}{n+1}\) will become an integer. Now find out the factors of 12 and try to build up some logic how to make this \(\frac{12}{n+1}\) an integer.

Step 4

So you can easily say that the factors of 12 are 1,2,3,4,6 and 12 respectively now try to think who you can use this information here in this \(\frac{12}{n+1}\). Like what are the values of n (from the factors of 12) in order to make it a (n+1) factor of 12.

 

 

 

Step 5 .

Here n can take values 0,1,2,3,5 and 11 respectively as n+1 must be a factor of  12 . But here 0 is not a positive integer so you have to exclude 0 so you are left with 5 different values of n . So your answer is 5

 

Start with hints

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Themes in Mathematics

Why is the number zero most important in mathematics?

Zero is the smallest number non-negative integer the immediately precedes 1. It is an even number as it as it is divisible by 2 with the remainder itself 0 { 0 ≡ 0 (mod 2)} i.e. no remainder . It cannot be termed as a positive or a negative number. The correct way of describing zero would be a number which equals to cardinality or an amount of null size.

1 is the natural number that follows 0 and there is no natural number that precedes 0. 0 is usually not considered as a natural number but it is definitely an integer and therefore a rational and real number. It also falls under the category of complex and algebraic numbers.

0 is usually presented as the central number in a number line. 0 can definitely not be termed as a prime number as it has a number of factors and cannot be composite as well. The reason behind 0 not been termed as a composite number is the inability to express the digit as a product of prime numbers as 0 is itself a factor.

In the field of Mathematics, there are some basic rules for working with the number 0.

Let x be any real or complex number

Subtraction can be done in 2 ways

x-0=x (positive number)

0-x= -x (negative number)

In addition 0 is the identity element i.e.

x+0=x

on adding any number with o the result is the number itself.

Division again yields different results

\(\frac{0}{x}\)=0

but \(\frac{x}{0}\)= undefined, as no real number multiplied by 0 produces 1 thus 0 does not contain any multiplicative inverse.

Multiplication of any number with 0 yields 0

x.0=0

and,

\(\frac{0}{0}\)=0 ,this expression is expressed in order to find the limit of the indeterminate form \(\frac{f(x)}{g(x)}\).This is called the indeterminate form.This implies that if the limit of \(\frac{f(x)}{g(x)}\) exists then it can be solved using L’Hospital’s rule.

Another very interesting fact about 0 is that 0! yields 1. It is an exceptional case of empty product.

0 is also used in propositional statements where it usually represents true or false depending upon a specific condition. It is also denoted as a zero element for addition and if defined, then zero is denoted an absorbing element for multiplication in the filed of abstract algebra.

It has several other applications in set theory ( where it is represented as the lowest ordinal number), lattice theory where zero is represented as the bottom element of a lattice(bounded), category theory and recursive theory as well. In category theory zero represents a initial value or object of a category.

Categories
Themes in Mathematics

How did Aryabhatta invent zero? How did he get this idea? Why did he give zero an oval shape?

Aryabhata was one of the major Mathematician-Astronomers belonging to the classical age of Indian Astronomy and Mathematics. Born in Pataliputra,Magadha, he is regarded as one of the greatest Mathematician of all time. His famous works include the ‘Aryabhatiya’ whose Mathematical parts consists of topics on algebra, trigonometry and arithmetic, continued fractions, sum of power series, quadratic equations and sine tables.

One of his discoveries is the approximation of pi which is given by him in Aryabhatia,

“Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.”

The calculation is obtained as 3.1416 which is close to the actual value of \(\pi\)(3.14159).

Before going to Aryabhata’s invention of zero lets know a little bit about the Indian History of number zero.

Acharya Pingala, a Sanskrit scholar and an Indian Mathematician first used the Sanskrit word ‘Sunya’, referred to as Zero.The word ‘Sunya’ means void or empty. It is believed that the first text to use the decimal place value system(includes zero) was first used in Jain text or Cosmology named ‘Lokavibhaga’ . This is where the term ‘Sunya’ was used.

‘Bakshali Manuscript’, an Arithmetic manual on merchants records the symbol of zero which is a dot like structure having a hollow structure signifying void or nothing..These manuscripts were brought up by Radiocarbon dating ( which is a method of determining the age of an object using radiocarbon) in 2017. The ages were recorded to come from 224-383 AD, 680-779 AD, and 885-993 AD. This marks the world’s oldest record of the application of the symbol of Zero.

In Mathematics there is a term called the Decimal place Value System also called Positional Notation. This means that the value of a number is determined by the position of the digit that is the value of a number is actually the product of the digit by a factor which is determined by the position of the digit.

For example lets take three identical digits 999. Here the interesting part is in words the number is written as nine hundred and ninety nine . The hundreds tens and the units here are being determined by the position of the digits that is digit at the first place represents the units, second place represents the tens and the third place represents hundreds. Similarly any digit at the fourth place shall reprimand thousands.

This concept of the place value system, although was first used in ‘Bakshali Manuscript’ held a very important place in Aryabhata’s work. But the symbol for Zero was not used by Aryabhata. The use of Zero as a ‘digit’ was first used in India during the Gupta Period.

George Ifrah, a French Mathematician stated that the concept and understanding of zero as a ‘digit’ was first given by Aryabhata in his place value system because the counting system of digits is not possible without the place value system or zero. Also calculation performed by Aryabhata on square and cubic roots cannot be done if the numbers are not arranged in accordance with the place value system or zero. This concept of Zero is considered to be one of the best and greatest achievements of Indian Mathematics.

Now the rules for using Zero as a digit was first introduced in Brahmasputha Siddhanta, by Bramhagupta whereas in some stances his rules differ from the modern rules, one being on dividing zero by zero the result yields zero.

Categories
AMC 10 USA Math Olympiad

AMC 10A Year 2005 Problem 22 Sequential Hints

Understand the problem

Let S be the set of the 2005 smallest positive multiples of 4, and let T be the set of the 2005 smallest positive multiples of 6. How many elements are common to S and T?

(a) 166.       (b)333.      (c)500.      (d)668.      (e)1001

Source of the problem
American Mathematical Contest 2005 10A Problem 22

Topic
Number Theory 

Difficulty Level
5/10

Suggested Book
Challenges and Thrills in Pre College Mathematics

Excursion Of Mathematics 

Do you really need a hint? Try it first!

Step 1.

First use the concept of lcm in order to find the type of common elements in both the sets.

Step 2.

After getting the gcd as 12 you can easily see that the common elements in S and T must be in the form 12k (Where k is positive integers) . Now try to find out the number of elements in S and T.

Step 3. 

Here lies the main concept of this problem as S has 8020 elements and T has 12030 (That is \(T>S\)) elements so you can see that many multiples of 12 are in T but not in S. As T has more elements than S so it’s obvious that multiples of 12 in S will definitely present in the set T but the converse is not true . So you have to only concentrate in the set S . Now find out the multiples of 12 in S. Think and give it a try!!!!!!!!!!

Step 4

Now you can see that as the lcm(4,6) is 12

In S you can see 4*3=12 so the set becomes 4,8,12,16,20,24,28,32,36,40,…… .Now you can see the 3rd , 6th , 9th , …. element are multiples of 12 ,so the multiples of 12 in S are in 3p ( values of p will determine the position of multiples of 12 in set S)  positions. Now here comes the interesting part of the problem. Think carefully what we can do with this information and hint!!!!!!

Step 5 .

So now you know that every 3rd element in S will be a multiple of 12 .So you have to calculate [\(\frac{2005}{3}\)] where [.] is the greatest integer function (Which means as many time this 3rd position or the 3p form appears in the set you will get a multiple of 12 so we will use the greatest integer function in order to find out the multiples of 12 in S).

Therefore  [\(\frac{2005}{3}\)]=668. which is your required answer 

 

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