Find all positive integers $n$ such that $n^4 − 1$ is divisible by 5.
107725
Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
100336
Find an integer having the remainders $2,3,4,5$ when divided by $3,4,5,6$, respectively.
26087
Find all natural numbers $n > 1$ and $m > 1$ such that $1!3!5!\cdots(2n - 1)! = m!$
24053
How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?
77816
Proof problem: Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer.
75166
Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
72389
Positive integer $n$ such that $2n+1$ , $3n+1$ are both perfect squares
39716
Prove that $8(5)^n + 6(3)^{2n} - 80n - 14$ is divisible by $512$ for all $n \in \mathbb{N}$ by induction.
96268
Prove that $5$ divides $3^{3n+1}+2^{n+1}$
101927
Induction: Prove that $4^{n+1}+5^{2 n - 1}$ is divisible by 21 for all $n \geq 1$.
148906
Finding integers $m,n$ such that $(m-n)^2=4mn/(m+n-1)$
15268
Find all $x,y \in \mathbb N^*$ such that $\frac{1}{x} + \frac{1}{y}=5$
52032
Prove that ${n^5 - n}$ is divisible by 5
116363
If $P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$ , then prove that $P(x)$ is divisible by $x-1$
27798
Prove that $ n^3 + 5n$ is divisible by 6 for all $n\in \textbf{N}$
1507
The product of $n$ consecutive integers is divisible by $n$ factorial
19701
Prove $5\mid2^{4n}-1$ by induction