QUESTION 176
The number of prime numbers between 10 and 30 is
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Let $f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0$ be a polynomial with integer coefficients and whose degree is at least 2. Suppose each $a_i$ ($0 \leq i \leq n-1$) is of the form $$a_i = \pm \frac{17!}{r!(17-r)!}$$ with $1 \leq r \leq 16$. Show that $f(m)$ is not equal to zero for any integer $m$.

Let $f ( x )$ be a polynomial with integer coefficients such that for each nonnegative integer $n , f ( n ) = \mathrm { a }$ perfect power of a prime number, i.e., of the form $p ^ { k }$, where $p$ is prime and $k$ a positive integer. ($p$ and $k$ can vary with $n$.) Show that $f$ must be a constant polynomial using the following steps or otherwise. a) If such a polynomial $f ( x )$ exists, then there is a polynomial $g ( x )$ with integer coefficients such that for each nonnegative integer $n , g ( n ) =$ a perfect power of a fixed prime number. b) Show that a polynomial $g ( x )$ as in part a must be constant.

(a) Let $f \in \mathbb { Z } [ x ]$ be a non-constant polynomial with integer coefficients. Show that as $a$ varies over the integers, the set of divisors of $f ( a )$ includes infinitely many different primes.
(b) Assume known the following result: If $G$ is a finite group of order $n$ such that for integer $d > 0$, $d \mid n$, there is no more than one subgroup of $G$ of order $d$, then $G$ is cyclic. Using this (or otherwise) prove that the multiplicative group of units in any finite field is cyclic.

A positive integer $n$ is called a magic number if it has the following property: if $a$ and $b$ are two positive numbers that are not coprime to $n$ then $a + b$ is also not coprime to $n$. For example, 2 is a magic number, because sum of any two even numbers is also even. Which of the following are magic numbers? Write your answers as a sequence of four letters (Y for Yes and N for No) in correct order.
(i) 129
(ii) 128
(iii) 127
(iv) 100.

Find all pairs $(p, n)$ of positive integers where $p$ is a prime number and $p^{3} - p = n^{7} - n^{3}$.

Find all prime numbers $p$ such that $p ^ { 2 } + 2$ is also a prime number.

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.

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

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

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

Let $n \geqslant 1$. The maximum possible number of primes in the set $\{n+6, n+7, \ldots, n+34, n+35\}$ is
(A) 7
(B) 8
(C) 12
(D) 13

Let $R$ be a relation from the set $\{ 1,2,3 \ldots\ldots . , 60 \}$ to itself such that $R = \{ ( a , b ) : b = p q$, where $p , q \geq 3$ are prime numbers\}. Then, the number of elements in $R$ is
(1) 600
(2) 660
(3) 540
(4) 720

A two-digit number $AB$ is called a symmetric prime if both $AB$ and $BA$ are prime numbers.
For a symmetric prime number $AB$, which of the following cannot be the product A.B?
A) 7
B) 9
C) 15
D) 21
E) 63

Let a be a positive integer and $p = a^{2} + 5$. If p is a prime number, which of the following statements are true?
I. a is an even number. II. The remainder when p is divided by 4 is 1. III. $\mathrm{p} - 6$ is prime.
A) I and III B) Only I C) I and II D) Only III E) I, II and III

a, b are positive integers, p is a prime number and
$$a ^ { 3 } - b ^ { 3 } = p$$
Given this, which of the following is the equivalent of $a ^ { 2 } + b ^ { 2 }$ in terms of $p$?
A) $\frac { p + 1 } { 2 }$
B) $\frac { p + 3 } { 2 }$
C) $\frac { p + 2 } { 3 }$
D) $\frac { 2 p - 1 } { 2 }$
E) $\frac { 2 p + 1 } { 3 }$

Let $p , q , r$ be prime numbers with
$$2 < p < q < r < 15$$
Accordingly, how many different values can the product $p \cdot q \cdot r$ take?
A) 4
B) 6
C) 8
D) 10
E) 12

Let n be an integer greater than 2, and let the largest prime divisor of n be denoted by $\tilde{n}$. The terms of the sequence $(a_n)$ are defined for $n \geq 2$ as
$$a _ { n } = \left\{ \begin{aligned} 1 & , \tilde{n} < 10 \\ - 1 & , \tilde{n} > 10 \end{aligned} \right.$$
Accordingly, what is the sum $\sum _ { n = 15 } ^ { 30 } a _ { n }$?
A) 2
B) 3
C) 4
D) 5
E) 6

Let $x$, $y$ and $z$ be distinct prime numbers,
$$\begin{aligned} & x ( z - y ) = 18 \\ & y ( z - x ) = 40 \end{aligned}$$
the equalities are given.
Accordingly, what is the sum $\mathbf { x } + \mathbf { y } + \mathbf { z }$?
A) 17 B) 19 C) 21 D) 23 E) 25

Let $a$, $b$, and $c$ be prime numbers,
$$a ( a + b ) = c ( c - b ) = 143$$
Given the equalities, accordingly, what is the sum $a + b + c$?
A) 22
B) 26
C) 30
D) 32

Let $p$ and $r$ be distinct prime numbers. The number $180 \cdot r$ is an integer multiple of the number $p$.
Accordingly, the prime number $p$ definitely divides which of the following numbers?
A) $12 \cdot r$
B) $18 \cdot r$
C) $20 \cdot r$
D) $30 \cdot r$
E) $45 \cdot r$

The sum of five distinct prime numbers equals 100, and their product equals a six-digit natural number ABCABC.
Accordingly, what is the sum $A + B + C$?
A) 8 B) 11 C) 14 D) 17 E) 20