The digit at the unit place of $$(1! - 2! + 3! - \ldots + 25!)^{(1! - 2! + 3! - \ldots + 25!)}$$ is
(a) 0
(b) 1
(c) 5
(d) 9

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 $2^a - 5^b \cdot 7^c = 1$.

The last digit of $( 2004 ) ^ { 5 }$ is
(A) 4
(B) 8
(C) 6
(D) 2

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

The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is
(A) 7
(B) 8
(C) 11
(D) 12

$N$ 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 $d_1, d_2, \ldots, d_k$ 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

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

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

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

If $n$ is a positive integer such that $8 n + 1$ is a perfect square, then
(a) $n$ must be odd
(b) $n$ cannot be a perfect square
(c) $2 n$ cannot be a perfect square
(d) none of the above.

If $n$ is a positive integer such that $8 n + 1$ is a perfect square, then
(a) $n$ must be odd
(b) $n$ cannot be a perfect square
(c) $2 n$ cannot be a perfect square
(d) none of the above.

The digit in the units' place of the number $1 ! + 2 ! + 3 ! + \ldots + 99 !$ is
(a) 3
(b) 0
(c) 1
(d) 7.

The digit in the units' place of the number $1 ! + 2 ! + 3 ! + \ldots + 99 !$ is
(a) 3
(b) 0
(c) 1
(d) 7.

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.

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.