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NMTC Number Theory Problems and Solutions

NMTC 2010 Primary Stage 1 Question 1

$\mathrm{n}, \mathrm{a}$ are natural numbers each greater than 1 . If $a+a+a+a+\ldots+a=2010$, and there are $n$ terms on the left hand side, then the number of ordered pairs $(a, n)$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

14

NMTC 2019 Inter Stage 1 Question 17

The number of times the digit occurs in the result of $1+11+111+\ldots .+111$
111 (100digits) is $\ldots \ldots .$

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

11

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, $1+2+2+3+3+2+1$ and $1+2+3+4+3+2+1$ are cool sums. The minimum number of terms required to write 2019 as a cool sum is ……

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

89

NMTC 2019 Inter Stage 1 Question 25

For each positive integer $n$ let $f(n)=n^{4}-3 n^{2}+9$. Then the sum of all $f(n)$ which are prime is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

20

NMTC 2019 Inter Stage 1 Question 30

The product of four positive integers $a, b, c$ and $d$ is 9 ! The number $a, b, c, d$ satisfy $a b+a+b=$ 1224, $b c+b+c=549$ and $c d+c+d=351$. The $a+b+c+d=\ldots \ldots$

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

108

NMTC 2019 Junior Stage 1 Question 1

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

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

128

NMTC 2019 Junior Stage 1 Question 6

In the sequence $1,4,8,10,16,21,25,30$ and 43 , the number of blocks of consecutive terms whose sums are divisible by 11 is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

exactly four

NMTC 2019 Junior Stage 1 Question 10

In the subtraction below, what is the sum of the digits in the result ? $111 \ldots \ldots \ldots . .111$ (100 digits) $-222 \ldots . . .222$ (50 digits)

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

450

NMTC 2019 Junior Stage 1 Question 26

The least odd prime factor of $2019^{8}+1$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

97

NMTC 2019 Primary Stage 1 Question 27

Let $a, b, c$ be positive integers each less than 50 , such that $a^{2}-b^{2}=100 c$. The number of such triples $(a, b, c)$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

25

NMTC 2019 Sub Junior Stage 1 Question 1

If $4921 \times D=A B B B D$, then the sum of the digits of $A B B B D \times D$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

19

NMTC 2019 Primary Stage 1 Question 2

What is the $2019^{\text {th }}$ digit to the right of the decimal point, in the decimal representation of $\frac{5}{28}$ ?

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

8.

NMTC 2019 Sub Junior Stage 1 Question 3

If $\mathrm{X}$ is a 1000 digit number, $Y$ is the sum of its digits, $Z$ the sum of the digits of $Y$ and $W$ the sum of the digits of $Z$, then the maximum possible value of $W$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

11

NMTC 2019 Sub Junior Stage 1 Question 4

Let $x$ be the number $0.000$. 001 which has 2019 zeroes after the decimal point. Then which of the following numbers is the greatest?

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

$\frac{1}{x^{2}}$

NMTC 2019 Sub Junior Stage 1 Question 5

$A B C$
then the number of possible values for $A, B, C, D, E$ satisfying this equation where
If $\frac{C \mathrm{CBA}}{\mathrm{DEDD}}$ then the number of $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ and $\mathrm{E}$ are distinct digits is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

4

NMTC 2019 Sub Junior Stage 1 Question 9

$\mathrm{R} A \mathrm{~T}$
If $+\mathrm{MAC}$ and each alphabet represents a different digit, what is the maximum possible value $+\frac{\mathrm{VAT}}{\mathrm{FLAT}}$
of FLAT?

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

2450

NMTC 2019 Sub Junior Stage 1 Question 13

There are exactly 5 prime numbers between 2000 and 2030 . Note: $2021=43 \times 47$ is not a prime number. The difference between the largest and the smallest among these is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

26

NMTC 2019 Sub Junior Stage 1 Question 23

A teacher asks 10 of her students to guess her age. They guessed it as $34,38,40,42,46,48,51$, 54,57 and 59 . Teacher said "At least half of you guessed it too low and two of you are off by one. Also my age is a prime number". The teacher's age is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

47

NMTC 2019 Sub Junior Stage 1 Question 24

The sum of 8 positive integers is 22 and their LCM is 9 . The number of integers among these that are less than 4 is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

7

NMTC 2019 Sub Junior Stage 1 Question 25

The number of natural numbers $n \leq 2019$ such that $\sqrt[3]{48 n}$ is an integer is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

3

NMTC 2019 Sub Junior Stage 1 Question 30

The number of perfect cubes that lie between $2^{9}+1$ and $2^{18}+1$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

56

NMTC 2010 Primary Stage 1 Question 1

$\mathrm{n}, \mathrm{a}$ are natural numbers each greater than 1 . If $a+a+a+a+\ldots+a=2010$, and there are $n$ terms on the left hand side, then the number of ordered pairs $(a, n)$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

14

NMTC 2019 Inter Stage 1 Question 17

The number of times the digit occurs in the result of $1+11+111+\ldots .+111$
111 (100digits) is $\ldots \ldots .$

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

11

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, $1+2+2+3+3+2+1$ and $1+2+3+4+3+2+1$ are cool sums. The minimum number of terms required to write 2019 as a cool sum is ……

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

89

NMTC 2019 Inter Stage 1 Question 25

For each positive integer $n$ let $f(n)=n^{4}-3 n^{2}+9$. Then the sum of all $f(n)$ which are prime is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

20

NMTC 2019 Inter Stage 1 Question 30

The product of four positive integers $a, b, c$ and $d$ is 9 ! The number $a, b, c, d$ satisfy $a b+a+b=$ 1224, $b c+b+c=549$ and $c d+c+d=351$. The $a+b+c+d=\ldots \ldots$

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

108

NMTC 2019 Junior Stage 1 Question 1

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

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

128

NMTC 2019 Junior Stage 1 Question 6

In the sequence $1,4,8,10,16,21,25,30$ and 43 , the number of blocks of consecutive terms whose sums are divisible by 11 is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

exactly four

NMTC 2019 Junior Stage 1 Question 10

In the subtraction below, what is the sum of the digits in the result ? $111 \ldots \ldots \ldots . .111$ (100 digits) $-222 \ldots . . .222$ (50 digits)

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

450

NMTC 2019 Junior Stage 1 Question 26

The least odd prime factor of $2019^{8}+1$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

97

NMTC 2019 Primary Stage 1 Question 27

Let $a, b, c$ be positive integers each less than 50 , such that $a^{2}-b^{2}=100 c$. The number of such triples $(a, b, c)$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

25

NMTC 2019 Sub Junior Stage 1 Question 1

If $4921 \times D=A B B B D$, then the sum of the digits of $A B B B D \times D$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

19

NMTC 2019 Primary Stage 1 Question 2

What is the $2019^{\text {th }}$ digit to the right of the decimal point, in the decimal representation of $\frac{5}{28}$ ?

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

8.

NMTC 2019 Sub Junior Stage 1 Question 3

If $\mathrm{X}$ is a 1000 digit number, $Y$ is the sum of its digits, $Z$ the sum of the digits of $Y$ and $W$ the sum of the digits of $Z$, then the maximum possible value of $W$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

11

NMTC 2019 Sub Junior Stage 1 Question 4

Let $x$ be the number $0.000$. 001 which has 2019 zeroes after the decimal point. Then which of the following numbers is the greatest?

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

$\frac{1}{x^{2}}$

NMTC 2019 Sub Junior Stage 1 Question 5

$A B C$
then the number of possible values for $A, B, C, D, E$ satisfying this equation where
If $\frac{C \mathrm{CBA}}{\mathrm{DEDD}}$ then the number of $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ and $\mathrm{E}$ are distinct digits is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

4

NMTC 2019 Sub Junior Stage 1 Question 9

$\mathrm{R} A \mathrm{~T}$
If $+\mathrm{MAC}$ and each alphabet represents a different digit, what is the maximum possible value $+\frac{\mathrm{VAT}}{\mathrm{FLAT}}$
of FLAT?

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

2450

NMTC 2019 Sub Junior Stage 1 Question 13

There are exactly 5 prime numbers between 2000 and 2030 . Note: $2021=43 \times 47$ is not a prime number. The difference between the largest and the smallest among these is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

26

NMTC 2019 Sub Junior Stage 1 Question 23

A teacher asks 10 of her students to guess her age. They guessed it as $34,38,40,42,46,48,51$, 54,57 and 59 . Teacher said "At least half of you guessed it too low and two of you are off by one. Also my age is a prime number". The teacher's age is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

47

NMTC 2019 Sub Junior Stage 1 Question 24

The sum of 8 positive integers is 22 and their LCM is 9 . The number of integers among these that are less than 4 is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

7

NMTC 2019 Sub Junior Stage 1 Question 25

The number of natural numbers $n \leq 2019$ such that $\sqrt[3]{48 n}$ is an integer is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

3

NMTC 2019 Sub Junior Stage 1 Question 30

The number of perfect cubes that lie between $2^{9}+1$ and $2^{18}+1$ is

Hints and solutions are coming up soon.

Hints and solutions are coming up soon.

56

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