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.
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 $R$ 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 positive integer is called a squaresum if and only if it can be written as the sum of the squares of two integers. For example, 61 and 9 are both squaresums since $61 = 5 ^ { 2 } + 6 ^ { 2 }$ and $9 = 3 ^ { 2 } + 0 ^ { 2 }$. A prime number is called awkward if and only if it has a remainder of 3 when divided by 4 . For example, 23 is awkward since $23 = 5 \times 4 + 3$. A (true) theorem due to Fermat states that: A positive integer is a squaresum if and only if each of its awkward prime factors occurs to an even power in its prime factorisation. It follows that $5 \times 23 ^ { 2 }$ is a squaresum, since 23 occurs to the power 2 , but $5 \times 23 ^ { 3 }$ is not, since 23 occurs to the power 3 . Which one of the following statements is not true?
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