Prove these numbers are composite: $500^{27}-123^{12}$, $12^{75}+21^{75}$, $58^{32}-49^{32}$
9846
Prove $n-m \mid n^r - m^r\,$ [Factor Theorem, monomial case]
91285
Prove $a+b+c+d $ is composite
133017
Prove that $n^n>1\times3\times5\times7\times \dots\times(2n-1)$
66016
Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$
97303
Calculate the multiplicative inverse modulo a composite number
153053
Find numbers $a, b, c$ given that $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$
96027
Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite.
111267
Prove that: $2^a+3^b<3a+4b$
9319
If $n\ne 4$ is composite, then $n$ divides $(n-1)!$.
8364
To show that Fermat number $F_{5}$ is divisible by $641$.
27963
How many ways a composite number can be resolved into two factors which are prime to each other?
44705
Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$
49501
Prove that : $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ divisible by $12$
46556
How to prove $3^{\log_4n} = n^{\log_43}$?
143927
Binomial coefficients that are powers of 2
158807
Proving a fact: $\tan(6^{\circ}) \tan(42^{\circ})= \tan(12^{\circ})\tan(24^{\circ})$
76282
Prime number expressible as $2^2+3^3+5^5+7^7+11^{11}+\cdots$