The total number of two digit numbers n, such that 3^n+7^n is a multiple of 10, is Answer:…

Question

The total number of two digit numbers $n$, such that $3^n+7^n$ is a multiple of 10, is

Answer: 45

Accepted Answer

$7^n = (10-3)^n \equiv (-3)^n \pmod{10}$, so $3^n+7^n \equiv 3^n+(-3)^n \pmod{10}$.

If $n$ is even: $3^n+(-3)^n = 2\cdot3^n$, and $3^n \pmod{10}$ cycles $3,9,7,1$; this is never $\equiv0\pmod5$ pattern giving $2\cdot3^n
ot\equiv0\pmod{10}$ generally.

If $n$ is odd: $3^n+(-3)^n = 0$, always a multiple of 10.

So all odd two-digit $n$ (i.e. $11,13,\ldots,99$) work: $45$ values.

Solutions · 1