Prove that for all natural numbers $k$, $k^5-k$ is a multiple of $10$.
73307
Why is the last digit of $n^5$ equal to the last digit of $n$?
147485
A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.
28749
Proving that any common multiple of two numbers is a multiple of their least common multiple
76032
Prove that if m is a square integer then m is neither congruent to 2 modulo 5 nor congruent to 3 modulo 5
96903
How to show via inductive proof (and without the use of the binomial theorem) that $ {n\choose k} \in \mathbb{N} $?
83738
Prove $(k,n+k)=1$ iff $(k,n)=1$
63416
Prove $(n^5-n)$ is divisible by 5 by induction.
120227
Proving $6^n - 1$ is always divisible by $5$ by induction
65757
Prove that $n! + k$ is a composite number
139935
Show that any set of 7 distinct integers includes two integers $x$ and $y$, such that either $x-y$ or $x+y$ is divisible by 10
57746
Prove that, for any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms.
43010
Prove that for every n ∈ Z, there exist a, b ∈ Z such that $n = 5a + 2b$.
31766
Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$
3611
How to prove $n^5 - n$ is divisible by $30$ without reduction
24403
Prove that $ n < 2^{n}$ for all natural numbers $n$.
142534
Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
43567
Some hints for "If a prime $p = n^2+5$, then $p\equiv 1\mod 10$ or $p\equiv 9\mod 10$"