The power of exponents

Algebra Level 2

How many positive integers $n$ , such that $n$ is divisible by $3$ , can be expressed in the form

$\large 2^m + 5^{m-1} + 7^{m-2}$

where $m$ is a positive integer larger than $1$ ?

Note: If the answer is infinty, enter $-1$ as an input.

The answer is 0.

1 solution

Chris Lewis
Dec 16, 2019

There are no such numbers. Let $a_m=2^m+5^{m-1}+7^{m-2}$ . Working modulo $3$ , whenever $m$ is even, we have $2^m \equiv 1$ , $5^{m-1} \equiv 2$ , and $7^{m-2} \equiv 1$ , so if $m$ is even, $a_m \equiv 1$ , ie it leaves a remainder of $1$ when divided by $3$ .

If $m$ is odd, $2^m \equiv 2$ , $5^{m-1} \equiv 1$ , and $7^{m-2} \equiv 1$ , so again $a_m \equiv 1$ .

The power of exponents

The answer is 0.

1 solution

0 pending reports