Practice (112)

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Let $P(x)$ be a polynomial with integer coefficients satisfying that both $P(0)$ and $P(1)$ are odd. Show that $P(x)$ has no integer zeros.

If $2016$ consecutive integers are added together, where the $999^{th}$ number in the sequence is $1,244,584$, what is the remainder when this sum is divided by $6$?

Prove that $7\mid 8^n-1$ for $n\ge 1$.


Show that $5\mid 4^{2n}-1$ for $n\ge 1$.


Prove that $15\mid 4^{2n}-1$ for $n\ge 1$.


An integer $N$ is selected at random in the range $1\leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?

Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$, the number of divisors of $k p+1$ between $k$ and $p$ exclusive is $a_k$. Find the value of $a_1+a_2+\ldots + a_{p-1}$.

Does there exist a polynomial $P(x)$ such that $P(1)=2015$ and $P(2015)=2016$?

Let $n$ be any positive integer, show that $$(5n+1)(5n+2)(5n+3)(5n+4)\equiv -1 \pmod{25}$$


Let $p$ be an odd prime. Show that $$\sum_{j=0}^p\binom{p}{j}\binom{p+j}{j}\equiv 2^p +1 \pmod{p^2}$$


Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered.


How many positive integers not exceeding $100$ are there such that the value of $(3^x-x^2)$ is a multiple of $5$?

Let $\mathbb{S}$ be the set of integers between $1$ and $2^{40}$ that contain two $1$s when written in base $2$. What is the probability that a random integer from $\mathbb{S}$ is divisible by $9$?


Find the multiplicative order of $3$ modulo $17$.


Find the multiplicative order of $5$ modulo $19$.


Show that if integer $a$ has multiplicative order of $hk$ modulo $n$, then $a^h$ has order of $k$ modulo $n$.


Let $p$ be an odd prime, and integer $a$ has multiplicative order of $2k$ modulo $p$, then $a^k\equiv -1\pmod{p}$.


Let $n$ be an odd integer greater than $1$, then $n$ is the multiplicative order of $2$ modulo $(2^n-1)$.


(Thue's theorem) Let $p$ be a prime. Show that for any integer $a$ such that $p\not\mid a$, there exist positive integers $x$, $y$ not exceeding $\lfloor{\sqrt{p}}\rfloor$ satisfying $ax\equiv y\pmod{p}$ or $ax\equiv -y\pmod{p}$.


Show that if the equation $a^2 + 1\equiv 0\pmod{p}$ is solvable for some $a$, then $p$ can be represented as a sum of two squares.


Show that a prime $p > 2$ is a sum of two squares if and only if $p\equiv 1\pmod{4}$.


(Two Squares Theorem) Show that a positive integer $n$ is a sum of two squares if and only if each prime factor $p$ of $n$ such that $p\equiv 3\pmod{4}$ occurs to an even power in the prime factorization of $n$.


Find the smallest integer $N$ such that $\varphi(n) \ge 5$ holds for all integer $n \ge N$.


Show that two positive integers $m$ and $n$ are co-prime if and only if $\varphi(mn)=\varphi(m)\varphi(n)$.


Let $n > 4$ be a composite number. Show that $(n-1)!\equiv 0\pmod{n}$.