2026/017) Let m and n be positive integers such that 5 divides $2^n+3^m$. Prove that 5 divides $2^m+3^n$

 

Because 5 is a small number we can work as below each exponent

Working in mod 5 we have

$,2^1 = 2,2^2= 4, 2^3 = 8 = 3,2^4=1$

$3^1=3, 3^2 = 9 = 4 , 3^3 = 27 = 2,3^4=1$

so $2^4 + 3^2$ = 0 mod 5 also $3^4 +2^2 = 0$ mod 5

$2^1+ 3^1 = 0$ mod 5 ( n=m)

$2^3 + 3^3 = 0$ mod 5(n=-m)

We have checked for all combinations that it is true