Consider a right-angled triangle with integer-valued sides $a < b < c$ where $a, b, c$ are pairwise co-prime. Let $d = c - b$. Suppose $d$ divides $a$. Then
(a) Prove that $d \leq 2$.
(b) Find all such triangles (i.e. all possible triplets $a, b, c$) with perimeter less than 100.

The number of ways one can express $2 ^ { 2 } 3 ^ { 3 } 5 ^ { 5 } 7 ^ { 7 }$ as a product of two numbers $a$ and $b$, where $\operatorname { gcd } ( a , b ) = 1$, and $1 < a < b$, is
(A) 5 .
(B) 6 .
(C) 7 .
(D) 8 .

Define $a = p ^ { 3 } + p ^ { 2 } + p + 11$ and $b = p ^ { 2 } + 1$, where $p$ is any prime number. Let $d = \operatorname { gcd } ( a , b )$. Then the set of possible values of $d$ is
(A) $\{ 1,2,5 \}$.
(B) $\{ 2,5,10 \}$.
(C) $\{ 1,5,10 \}$.
(D) $\{ 1,2,10 \}$.

Let $n > 1$ be the smallest composite integer that is coprime to $\frac{10000!}{9900!}$. Then
(A) $n \leqslant 100$
(B) $100 < n \leqslant 9900$
(C) $9900 < n \leqslant 10000$
(D) $n > 10000$

The sum of all natural numbers $n$ such that $100 < n < 200$ and H.C.F.$(91, n) > 1$ is
(1) 3203
(2) 3221
(3) 3121
(4) 3303

Let $A = \{ 2,3,4,5 , \ldots , 30 \}$ and $\sim$ be an equivalence relation on $A \times A$, defined by $( a , b ) \simeq ( c , d )$, if and only if $ad = bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $( 4,3 )$ is equal to :
(1) 5
(2) 6
(3) 8
(4) 7

If $\operatorname{gcd}(m, n) = 1$ and $1^{2} - 2^{2} + 3^{2} - 4^{2} + \ldots + (2021)^{2} - (2022)^{2} + (2023)^{2} = 1012m^{2}n$ then $m^{2} - n^{2}$ is equal to
(1) 240
(2) 200
(3) 220
(4) 180

Let $A = \{ 1,2,3 , \ldots , 10 \}$ and $B = \left\{ \frac { m } { n } : m , n \in A , m < n \right.$ and $\left. \operatorname { gcd } ( m , n ) = 1 \right\}$. Then $n ( B )$ is equal to:
(1) 36
(2) 31
(3) 37
(4) 29

Let $a$ and $b$ be natural numbers such that the greatest common divisor of $a$ and $b$ is 3. We are to find the natural numbers $a$ and $b$ such that
$$3 a - 2 b = \ell + 3$$
is satisfied, where $\ell$ is the least common multiple of $a$ and $b$.
When we set $a = 3 p$ and $b = 3 q$, the natural numbers $p$ and $q$ are mutually prime (co-prime), and hence $\ell = \mathbf { N } p q$. Thus using $p$ and $q$, the equality (1) can be transformed to
$$p q - \mathbf { O } p + \mathbf { P } q + \mathbf { Q } = 0 .$$
This can be further transformed to
$$( p + \mathbf { R } ) ( q - \mathbf { S } ) = - \mathbf { S } \mathbf { T } .$$
Among the pairs of integers $p$ and $q$ which satisfy this equation, the pair such that both $a$ and $b$ are natural numbers is
$$p = \mathbf { U } , \quad q = \mathbf { V } ,$$
which gives
$$a = \mathbf { W X } , \quad b = \mathbf { Y } .$$

On the set of positive integers, the operations $\oplus$ and $\otimes$ are defined using the greatest common divisor and least common multiple as follows:
$$\begin{aligned} & a \oplus b = \operatorname { GCD } ( a , b ) \\ & a \otimes b = \operatorname { LCM } ( a , b ) \end{aligned}$$
Accordingly, what is the result of the operation $18 \oplus ( 12 \otimes 4 )$?
A) 2
B) 3
C) 6
D) 8
E) 9

The least common multiple of $b$ and $40$ is $120$.
Accordingly, how many different positive integers $b$ are there?
A) 6
B) 8
C) 10
D) 12
E) 14

Let p and q be distinct prime numbers such that
$$\begin{aligned} & a = p ^ { 4 } \cdot q ^ { 2 } \\ & b = p ^ { 2 } \cdot q ^ { 3 } \end{aligned}$$
Which of the following is the greatest common divisor of numbers a and b?
A) $p ^ { 5 } \cdot q ^ { 4 }$
B) $p ^ { 4 } \cdot q ^ { 3 }$
C) $p ^ { 3 } \cdot q ^ { 4 }$
D) $p ^ { 2 } \cdot q ^ { 2 }$
E) $p ^ { 2 } \cdot q ^ { 3 }$

Let n be a positive integer, and let $S(n)$ denote the set of positive integers that divide n without remainder.
Accordingly, how many elements does the intersection set $S(60) \cap S(72)$ have?
A) 8 B) 9 C) 6 D) 5 E) 4

$\mathbf { a } < \mathbf { b } < \mathbf { c }$ are positive integers and
$$\begin{aligned} & \gcd ( a , b ) = 5 \\ & \gcd ( b , c ) = 4 \end{aligned}$$
Given this, what is the minimum value that the sum $\mathbf { a + b + c }$ can take?
A) 27
B) 35
C) 39
D) 45
E) 49

Let $a$ and $b$ be distinct positive integers such that LCM(a,b) equals a prime number.
Accordingly,\ I. $a$ and $b$ are coprime numbers.\ II. The sum $a + b$ is an odd number.\ III. The product $\mathrm{a} \cdot \mathrm{b}$ is an odd number.
Which of the following statements are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and II\ E) II and III

Let p, r, and t be different prime numbers;

• Integer multiples of p form set A,
• Integer multiples of r form set B,
• Integer multiples of t form set C.

It is known that two of the numbers 220, 245, 330, and 350 are elements of the blue-colored set, and the other two are elements of the yellow-colored set. Accordingly, what is the sum $\mathbf { p } + \mathbf { r } + \mathbf { t }$?
A) 10
B) 14
C) 15
D) 21
E) 23

Let $m$ and $n$ be positive integers such that
$$\begin{aligned} & \gcd ( m , n ) + \text{lcm} ( m , n ) = 289 \\ & m + n \neq 289 \end{aligned}$$
What is the sum $m + n$?
A) 41
B) 43
C) 45
D) 47
E) 49

Let $a, b, c$ and $d$ be positive integers. $$\begin{aligned} & M = 6^{a} \cdot 5^{b} \\ & N = 10^{c} \cdot 9^{d} \end{aligned}$$ For the numbers $M$ and $N$ $$\begin{aligned} & \gcd(M, N) = 2^{3} \cdot 3^{2} \cdot 5 \\ & \text{lcm}(M, N) = 2^{5} \cdot 3^{3} \cdot 5^{5} \end{aligned}$$ are given. Accordingly, what is the sum $a + b + c + d$?
A) 8 B) 9 C) 10 D) 11 E) 12