Practice (106)

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Show that $3^n-2$ is a square only for $n=1$ and $n=3$.

Solve in integers the equation $x^2+y^2+z^2-2xyz=0$

Find all positive integer $n$ such that $n$ is a square and its last four digits are the same.

Show the equation $x^2 + y^2-8z^3 = 6$ has no integer solution.

What is the last digit of $17^{17^{17^{17}}}$?

What is the smallest positive integer $n$ such that $20\equiv n^{15} \pmod{29}$?


Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3\le p<100$, and $1\le a < p$ such that the sum $a+a^2+a^3+ \cdots + a^{(p-2)!}$ is not divisible by $p$?


For every integer $n$, let $m$ denote the integer made up of the last four digit of $n^{2015}$. Consider all positive integer $n < 10000$, let $A$ be the number of cases when $n > m$, and $B$ be the number of cases when $n < m$. Compute $A-B$.

If for any integer $k\ne 27$ and $\big(a-k^{2015}\big)$ is divisible by $(27-k)$, what is the last two digits of $a$?

Solve $4x^2+27x-9\equiv 0\pmod{15}$

Solve $5x^3 -3x^2 +3x-1\equiv 0\pmod{11}$

Solve $3x^{15}-x^{13}-x^{12} -x^{11} -3x^5 +6x^3 -2x^2 +2x-1\equiv 0 \pmod{11}$

Solve $14x\equiv 30 \pmod{21}$

Solve $17x\equiv 229\pmod{1540}$.

Solve $$\left\{ \begin{array}{rcl} x &\equiv 2 &\pmod{3}\\ x &\equiv 2 &\pmod{5}\\ x &\equiv -3 &\pmod{7}\\x &\equiv -2 &\pmod{13} \end{array}\right.$$


What is the smallest positive integer greater than $5$ which leaves a remainder of $5$ when divided by each of $6$, $7$, $8$, and $9$?


Determine the units digit of the sum $0!+1!+2!+\cdots+n!+\cdots+20!$?

What is the units digit of $-1\times 2008 + 2 \times 2007 - 3\times 2006 + 4\times 2005 +\cdots-1003\times 1006 + 1004 \times 1005$?

Solve $$\left\{ \begin{array}{rcl} 4x & \equiv 14 &\pmod{15}\\ 9x & \equiv 11 &\pmod{20}\\ \end{array}\right.$$


Let sequence $g(n)$ satisfy $g(1)=0, g(2)=1, g(n+2)=g(n+1)+g(n)+1$ where $n\ge 1$. Show that if $n$ is a prime greater than 5, then $n\mid g(n)[g(n)+1]$.


Show that all the terms of the sequence $a_n=\frac{(2+\sqrt{3})^n-(2-\sqrt{3})^n}{2\sqrt{3}}$ are integers, and also find all the $n$ such that $3 \mid a_n$.

What is the remainder when $2021^{2020}$ is divided by $10^4$?


Find the least non-negative residue of $70! \pmod{5183}$.


Compute $50^{250} \pmod{83}$ .

What is the last digit of $7^{222}$?