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

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 $\ldots \ldots .$

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 ……

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

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 $a+b+c+d=\ldots \ldots$

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

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

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)

NMTC 2019 Junior Stage 1 Question 26

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

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

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

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 $\frac{5}{28}$ ?

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

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?

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

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?

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

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

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

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

NMTC 2019 Sub Junior Stage 1 Question 30

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

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

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 $\ldots \ldots .$

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 ……

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

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 $a+b+c+d=\ldots \ldots$

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

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

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)

NMTC 2019 Junior Stage 1 Question 26

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

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

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

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 $\frac{5}{28}$ ?

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

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?

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

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?

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

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

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

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

NMTC 2019 Sub Junior Stage 1 Question 30

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