Question

How will you show ? `5^n+1` is always divisible by 7 when n is a positive odd integer divisible by 3.

Answer 1

(see below for proof)

Explanation:

If `n` is a positive odd integer divisible by `3`
we can replace `n` with `3(2p+1)` for some `"odd "p inZZ`

`5^n+1`
`color(white)("XXX")= 5^(3(p))+1`

`color(white)("XXX")=125^(p)+1`

Noting that `125=7xx18-1`

we have
`5^n+1`
`color(white)("XXX")=(7xx18-1)^(p)+1`

`rArr(5^n+1)_("mod "7)= (-1)^(p)+1`

but `(-1)^("any odd integer") = -1`

`rArr (5^n+1)_("mod "7)=-1+1=0`

`rArr (5^n+1)` is divisible by `7`