Thread: n^4 not / 5 is of form 5n+1

1. n^4 not / 5 is of form 5n+1

After being out of university for 22 years, I have been asked for help with the following HS math team practice problem and I realize I have no idea how to proceed. Would anyone be able to shed some light? (Number theory was never my strong point)

"Prove that the 4th power of a number not divisible by 5 is of the form 5n+1"

Thanks!

2. Hope this helps

1 to the 4th power is 1.
2 to the 4th power is 16.
3 to the 4th power is 81
4 to the 4th power is 256
5 to the 4th power is 625

These are evenly divisible by 5 with a remainder of 1.
Observe that the remaining numbers from 6 to 9 raised to the 4th power match up with the digits in the one position of the first four I've already raised to the 4th power. The rest, 10 and above, are trivial. (PS I haven't been to college in over 30 years myself)

3. This is a specific case of Fermat's "little" theorem.

Are you familiar with modular arithmetic? If so then it should be fairly obvious that you only have to show it for $1^4, 2^4, 3^4, 4^4$. (Because $(5m+k)^4\equiv k^4 \mod 5$).