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# NMTC Number Theory Problems and Solutions

### NMTC 2010 Primary Stage 1 Question 1

are natural numbers each greater than 1 . If , and there are terms on the left hand side, then the number of ordered pairs is

Value of will be greater than . So first we can find out the factors of .

So,

When the value of is , value of is and when the value of is 3 then the value of n is and vice versa.

So the ordered pairs will be,

So the number of ordered pairs is .

### NMTC 2019 Inter Stage 1 Question 17

The number of times the digit occurs in the result of
111 (100digits) is

sum is

there are brackets
so comes times

### NMTC 2019 Inter Stage 1 Question 20

Let us call a sum of integers a cool sum if the first and last terms are 1 and each term differs from its neighbours by at most. For example, and are cool sums. The minimum number of terms required to write 2019 as a cool sum is ……

Sequence is

So minimum number of terms is .

### NMTC 2019 Inter Stage 1 Question 25

For each positive integer let . Then the sum of all which are prime is

is prime

sum of values

### NMTC 2019 Inter Stage 1 Question 30

The product of four positive integers and is 9 ! The number satisfy 1224, and . The

given

Therefore
and

So

### NMTC 2019 Junior Stage 1 Question 1

The number of 6 digit numbers of the form "ABCABC", which are divisible by 13 , where and are distinct digits, and being even digits is

where Now and are even digits and are different digits .

Case-I: When is zero

Case-II : When is not zero

Total number of digits

Number possible

### NMTC 2019 Junior Stage 1 Question 6

In the sequence and 43 , the number of blocks of consecutive terms whose sums are divisible by 11 is

Exactly four

### NMTC 2019 Junior Stage 1 Question 10

In the subtraction below, what is the sum of the digits in the result ? (100 digits) (50 digits)

---------------------------------------------

49 times 1,49 times 8 and 1 times 0 and 9
Sum

Therefore

### NMTC 2019 Junior Stage 1 Question 26

The least odd prime factor of is

Let be an odd prime which divides
So

Now by Euler's theorem

So should be divisible by
Where is a prime
First two prime numbers which gives remainder when divided by is and
Case-1
(mode )

While
(mod )
So the answer is .

### NMTC 2019 Junior Stage 1 Question 27

Let be positive integers each less than 50 , such that . The number of such triples is

As is a multiple of

So it means the last digit of and is same.

So can be .

So there are such pairs .

One more pair for is .

So total pairs are possible.

### NMTC 2019 Sub Junior Stage 1 Question 1

If , then the sum of the digits of is

4-digit no. (4921) is multiplied by a single digit no. (D) \& result is five digit no., so definitely

So by hit \& trial we put the values of D from 3 to 9 .
at

So

Now ABBBD
Sum of digits

### NMTC 2019 Primary Stage 1 Question 2

What is the digit to the right of the decimal point, in the decimal representation of ?

[2 for pairs of 6 repeating number]

digit from right side to decimal is first digit in repetition

So correct answer is .

### NMTC 2019 Sub Junior Stage 1 Question 3

If is a 1000 digit number, is the sum of its digits, the sum of the digits of and the sum of the digits of , then the maximum possible value of is

digit no. If all digit are ' 9 ' so that maximum sum of digit of ' ' is 9000 So maximum value of is 9000

But for maximum sum of digit of is 35 for number (8999) So is maximum 35 .

Now for maximum sum of digit of is 11 for number So . Practical example: if Sum of digit of Sum of digit of Sum of digit of

### NMTC 2019 Sub Junior Stage 1 Question 4

Let be the number . 001 which has 2019 zeroes after the decimal point. Then which of the following numbers is the greatest?

(A) , (B) , (C) , (D)

From option

From option
From option

From option
So from options is greatest.

### NMTC 2019 Sub Junior Stage 1 Question 5

then the number of possible values for satisfying this equation where
If then the number of and are distinct digits is

_________

must be
Means or 11
Sum of is even is
So possible values of B is 0 or 5 .

But if we take as ' 0 ' so there is no carry forward \& Sum of , did not get different digit from D.
So B must be 5 .
Sum is convert into

_____

Now possible pairs of are
So total 4 possible solutions are there.

### NMTC 2019 Sub Junior Stage 1 Question 9

If and each alphabet represents a different digit, what is the maximum possible value
of FLAT?

T should be 0 or 5
But if we take as ' 5 ' sum is carry is forward and sum of 3 'A' and '1' never give unit digit 'A' So T must be '0'

Now again possible values of are but alphabet represents different digits so is 5 . For maximum value of FLAT, we take maximum value of as , but sum is is repeat.

So by taking as . We get maximum value of FLAT.

2029-2003=26

1+1+1+1+3+3+3+9=22

So numbers less than 4 is 7

### NMTC 2019 Sub Junior Stage 1 Question 25

The number of natural numbers such that is an integer is

So should be multiple of for integer value.
integer integer integer.

So should be multiple of for intege So, possible 'n' are

(reject)

So only three values of are possible

### NMTC 2019 Sub Junior Stage 1 Question 30

The number of perfect cubes that lie between and is

and and

Numbers lies between Total perfect cubes number .

### NMTC 2010 Primary Stage 1 Question 1

are natural numbers each greater than 1 . If , and there are terms on the left hand side, then the number of ordered pairs is

Value of will be greater than . So first we can find out the factors of .

So,

When the value of is , value of is and when the value of is 3 then the value of n is and vice versa.

So the ordered pairs will be,

So the number of ordered pairs is .

### NMTC 2019 Inter Stage 1 Question 17

The number of times the digit occurs in the result of
111 (100digits) is

sum is

there are brackets
so comes times

### NMTC 2019 Inter Stage 1 Question 20

Let us call a sum of integers a cool sum if the first and last terms are 1 and each term differs from its neighbours by at most. For example, and are cool sums. The minimum number of terms required to write 2019 as a cool sum is ……

Sequence is

So minimum number of terms is .

### NMTC 2019 Inter Stage 1 Question 25

For each positive integer let . Then the sum of all which are prime is

is prime

sum of values

### NMTC 2019 Inter Stage 1 Question 30

The product of four positive integers and is 9 ! The number satisfy 1224, and . The

given

Therefore
and

So

### NMTC 2019 Junior Stage 1 Question 1

The number of 6 digit numbers of the form "ABCABC", which are divisible by 13 , where and are distinct digits, and being even digits is

where Now and are even digits and are different digits .

Case-I: When is zero

Case-II : When is not zero

Total number of digits

Number possible

### NMTC 2019 Junior Stage 1 Question 6

In the sequence and 43 , the number of blocks of consecutive terms whose sums are divisible by 11 is

Exactly four

### NMTC 2019 Junior Stage 1 Question 10

In the subtraction below, what is the sum of the digits in the result ? (100 digits) (50 digits)

---------------------------------------------

49 times 1,49 times 8 and 1 times 0 and 9
Sum

Therefore

### NMTC 2019 Junior Stage 1 Question 26

The least odd prime factor of is

Let be an odd prime which divides
So

Now by Euler's theorem

So should be divisible by
Where is a prime
First two prime numbers which gives remainder when divided by is and
Case-1
(mode )

While
(mod )
So the answer is .

### NMTC 2019 Junior Stage 1 Question 27

Let be positive integers each less than 50 , such that . The number of such triples is

As is a multiple of

So it means the last digit of and is same.

So can be .

So there are such pairs .

One more pair for is .

So total pairs are possible.

### NMTC 2019 Sub Junior Stage 1 Question 1

If , then the sum of the digits of is

4-digit no. (4921) is multiplied by a single digit no. (D) \& result is five digit no., so definitely

So by hit \& trial we put the values of D from 3 to 9 .
at

So

Now ABBBD
Sum of digits

### NMTC 2019 Primary Stage 1 Question 2

What is the digit to the right of the decimal point, in the decimal representation of ?

[2 for pairs of 6 repeating number]

digit from right side to decimal is first digit in repetition

So correct answer is .

### NMTC 2019 Sub Junior Stage 1 Question 3

If is a 1000 digit number, is the sum of its digits, the sum of the digits of and the sum of the digits of , then the maximum possible value of is

digit no. If all digit are ' 9 ' so that maximum sum of digit of ' ' is 9000 So maximum value of is 9000

But for maximum sum of digit of is 35 for number (8999) So is maximum 35 .

Now for maximum sum of digit of is 11 for number So . Practical example: if Sum of digit of Sum of digit of Sum of digit of

### NMTC 2019 Sub Junior Stage 1 Question 4

Let be the number . 001 which has 2019 zeroes after the decimal point. Then which of the following numbers is the greatest?

(A) , (B) , (C) , (D)

From option

From option
From option

From option
So from options is greatest.

### NMTC 2019 Sub Junior Stage 1 Question 5

then the number of possible values for satisfying this equation where
If then the number of and are distinct digits is

_________

must be
Means or 11
Sum of is even is
So possible values of B is 0 or 5 .

But if we take as ' 0 ' so there is no carry forward \& Sum of , did not get different digit from D.
So B must be 5 .
Sum is convert into

_____

Now possible pairs of are
So total 4 possible solutions are there.

### NMTC 2019 Sub Junior Stage 1 Question 9

If and each alphabet represents a different digit, what is the maximum possible value
of FLAT?

T should be 0 or 5
But if we take as ' 5 ' sum is carry is forward and sum of 3 'A' and '1' never give unit digit 'A' So T must be '0'

Now again possible values of are but alphabet represents different digits so is 5 . For maximum value of FLAT, we take maximum value of as , but sum is is repeat.

So by taking as . We get maximum value of FLAT.

2029-2003=26

1+1+1+1+3+3+3+9=22

So numbers less than 4 is 7

### NMTC 2019 Sub Junior Stage 1 Question 25

The number of natural numbers such that is an integer is

So should be multiple of for integer value.
integer integer integer.

So should be multiple of for intege So, possible 'n' are

(reject)

So only three values of are possible

### NMTC 2019 Sub Junior Stage 1 Question 30

The number of perfect cubes that lie between and is

and and

Numbers lies between Total perfect cubes number .

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