Select Page

Competency in Focus: Number Theory

This problem from American Mathematics Contest 10B (AMC 10B, 2019) is based on calculation of number theory. It is Question no. 19 of the AMC 10B 2019 Problem series.

Next understand the problem

Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ $\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$
Source of the problem
American Mathematical Contest 2019, AMC 10B Problem 19

Number Theory

4/10
Suggested Book

Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

Do you really need a hint? Try it first!
Any number is divisible by all of its factors. For eaxmple 50 is divisible by $2,5,10$ and $25$ out of these their are some prime numbers called Prime factors.
The prime factor of 100,000 are only 2 and 5, the rest of them are not the prime factor, they are composite factor. Also The prime factorization of $100,000$ is $2^5 \cdot 5^5$.
Any Number which divides 100,000 must be multiple of 2 and (or) 5. So it can be 10=5×2 or $200=2^{3} 5^{2}$.
Since prime factorization of $100,000$ is $2^5 \cdot 5^5$. Thus We can find possible value of a,b,c and d being between 0 and 5.

AMC - AIME Program

AMC - AIME - USAMO Boot Camp for brilliant students. Use our exclusive one-on-one plus group class system to prepare for Math Olympiad

Arithmetic sequence | AMC 10A, 2015 | Problem 7

Try this beautiful problem from Algebra: Arithmetic sequence from AMC 10A, 2015, Problem. You may use sequential hints to solve the problem.

Problem based on Cylinder | AMC 10A, 2015 | Question 9

Try this beautiful problem from Mensuration: Problem based on Cylinder from AMC 10A, 2015. You may use sequential hints to solve the problem.

Median of numbers | AMC-10A, 2020 | Problem 11

Try this beautiful problem from Geometry based on Median of numbers from AMC 10A, 2020. You may use sequential hints to solve the problem.

Cubic Equation | AMC-10A, 2010 | Problem 21

Try this beautiful problem from Algebra, based on the Cubic Equation problem from AMC-10A, 2010. You may use sequential hints to solve the problem.

Problem on Fraction | AMC 10A, 2015 | Question 15

Try this beautiful Problem on Fraction from Algebra from AMC 10A, 2015. You may use sequential hints to solve the problem.

Side of Square | AMC 10A, 2013 | Problem 3

Try this beautiful problem from Geometry: Side of Square from AMC-10A (2013) Problem 3. You may use sequential hints to solve the problem.

Counting Days | AMC 10A, 2013 | Problem 17

Try this beautiful problem from Algebra based on Counting Days from AMC-10A (2013), Problem 17. You may use sequential hints to solve the problem.

Order Pair | AMC-10B, 2012 | Problem 10

Try this beautiful problem from Algebra, based on Order Pair problem from AMC-10B, 2012. You may use sequential hints to solve the problem

Chosing Program | AMC 10A, 2013 | Problem 7

Try this beautiful problem from Combinatorics based on Chosing Program from AMC-10A (2013), Problem 7. You may use sequential hints to solve the problem.

Triangle Area Problem | AMC-10A, 2009 | Problem 10

Try this beautiful problem from Geometry: The area of triangle AMC-10, 2009. You may use sequential hints to solve the problem