Prove that for $n\in\Bbb{Z}^+$, $n$ has an odd number of positive divisors if and only if $n$ is the square of an integer.
120498
Why perfect square has odd number of factors
7414
Prove that odd perfect square is congruent to $1$ modulo $8$
23879
If $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$
79219
Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
64654
Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
140294
prove that $n >2$ and $n$ is a prime then $n$ is odd
51952
Show that $x^{n-1}+\cdots +x+1$ is irreducible over $\mathbb Z$ if and only if $n$ is a prime.
142992
Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$
118452
how to prove $\mathbb Z_n$ has divisors of zero if and only if $n$ is not prime
16660
Prove that any positive number has an $n$th root
82071
divisor of a product of integers is a product of divisors
21840
Zero divisors and units for the ring $\mathbb{Z}/n\mathbb{Z}$ with integer $n$
90806
Find all positive integers $n$ such that $n$ is divisible by all the positive integers less than or equal to $\sqrt{n}$
157294
Prove that if $n>1$ is odd, then $1^n+2^n+3^n+\ldots+(n-1)^n$ is divisible by $n$
77564
How to prove for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$?
92166
Number of odd binomial coefficients is a power of $2$
48251
Prove that if $n \in \mathbb{N}$, then $\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$ where $d(n)$ is the divisor function.