132- The number of positive multiples of positive integer $x = 3^m \times 5^n$ that are also multiples of positive integer $\dfrac{x}{40}$ is more than 12. What is the minimum value of $x$?
(1) $640$ (2) $800$ (3) $1000$ (4) $1280$

133- What is the average of the largest and smallest three-digit numbers of the form $\overline{aba}$ that are multiples of 12?
(1) $348$ (2) $450$ (3) $570$ (4) $574$

134- If the integer part of dividing natural number $a > 9$ by 11 is 3 more than its remainder, what is the probability that $9 - a$ is divisible by 24?
(1) $\dfrac{13}{22}$ (2) $\dfrac{6}{11}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{5}{11}$
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135. If $m$ is the largest natural number such that $36 \equiv (m)! \pmod{15}$, then $m^{123}$ divided by $15$, the remainder is which of the following?
(1) $1$ (2) $2$ (3) $4$ (4) $6$

38. What is the smallest three-digit natural number $x$ that satisfies the equation $273 = 63x + 77y$?
(1) $4$ (2) $5$ (3) $8$ (4) $9$

Let $X = \{0,1,2,3,\ldots,99\}$. For $a, b$ in $X$, we define $a * b$ to be the remainder obtained by dividing the product $ab$ by 100. For example, $9 * 18 = 62$ and $7 * 5 = 35$. Let $x$ be an element in $X$. An element $y$ in $X$ is called the inverse of $x$ if $x * y = 1$. Find all elements of $X$ that have inverses and write down their inverses.

Let $g(n) = 5^k$ where $k$ is the number of distinct primes dividing $n$, and let $h(n) = 0$ if $n$ is divisible by $k^2$ for some integer $k > 1$, and $h(n) = 1$ otherwise.
a) Show that $g(mn) = g(m)g(n)$ does not hold in general, and determine when it holds.
b) Show that $h(mn) = h(m)h(n)$ for all positive integers $m, n$.

Show that $n^4 + 4^n$ is composite for all integers $n > 1$.

Let $n$ be an integer. The number of primes which divide both $n^{2}-1$ and $(n+1)^{2}-1$ is
(a) At most one.
(b) Exactly one.
(c) Exactly two.
(d) None of the above.

Among all the factors of $4 ^ { 6 } 6 ^ { 7 } 21 ^ { 8 }$, the number of factors which are perfect squares is
(a) 240
(b) 360
(c) 400
(d) 640

Find the number of integer solutions to $x^2 + y^2 = 2007$.

Find the last digit of $9! + 3^{9966}$.

Find the number of positive integer solutions to $2^a - 5^b \cdot 7^c = 1$.

Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a^n + b^n$ is prime, what can you say about $n$?
(A) The integer $n$ must be 2
(B) The integer $n$ need not be 2, but must be a power of 2
(C) The integer $n$ need not be a power of 2, but must be even
(D) None of the above is necessarily true

$N$ is a 50 digit number. All the digits except the 26th from the right are 1. If $N$ is divisible by 13, then the unknown digit is
(A) 1
(B) 3
(C) 7
(D) 9

If $n$ is a positive integer such that $8n + 1$ is a perfect square, then
(A) $n$ must be odd
(B) $n$ cannot be a perfect square
(C) $2n$ cannot be a perfect square
(D) none of the above

The digit in the unit's place of the number $1! + 2! + 3! + \ldots + 99!$ is
(A) 3
(B) 0
(C) 1
(D) 7

Let $P = \{2^a \cdot 3^b : 0 \leq a, b \leq 5\}$. Find the largest $n$ such that $2^n$ divides some element of $P$.
(A) $n = 20$ (B) $n = 22$ (C) $n = 24$ (D) $n = 26$

Let $a, b, c$ be positive integers with $a^2 + b^2 = c^2$. Which of the following must be true?
(A) 3 divides exactly one of $a, b$ (B) 3 divides $c$ (C) $3^3$ divides $abc$ (D) $3^4$ divides $abc$

Let $N$ be the smallest positive integer such that among any $N$ consecutive integers, at least one is coprime to $374 = 2 \times 11 \times 17$. Find $N$.
(A) 4 (B) 5 (C) 6 (D) 7

Find the number of ordered triples $(a, b, c)$ of positive integers such that $abc = 1000$.
(A) 90 (B) 100 (C) 110 (D) 120

Let $a _ { n } = \underbrace{1 \ldots 1}_{3^n \text{ digits}}$ with $3 ^ { n }$ digits. Prove that $a _ { n }$ is divisible by $3 a _ { n - 1 }$.

Let $a _ { n } = 1 \ldots 1$ with $3 ^ { n }$ digits. Prove that $a _ { n }$ is divisible by $3 a _ { n - 1 }$.

The last digit of $( 2004 ) ^ { 5 }$ is:
(a) 4
(b) 8
(c) 6
(d) 2

Let $d _ { 1 } , d _ { 2 } , \ldots , d _ { k }$ be all the factors of a positive integer $n$ including 1 and $n$. If $d _ { 1 } + d _ { 2 } + \ldots + d _ { k } = 72$, then $\frac { 1 } { d _ { 1 } } + \frac { 1 } { d _ { 2 } } + \cdots + \frac { 1 } { d _ { k } }$ is:
(a) $\frac { k ^ { 2 } } { 72 }$
(b) $\frac { 72 } { k }$
(c) $\frac { 72 } { n }$
(d) none of the above.