Question

Find all positive integers n such that n^4- 1 is divisible by 5.​

Answer 1

All positive integers except multiple of 5.

Explanation:

From the rule of divisibility, a number's last digit should be either 0 or 5 for it to be divisible by 5.

For
`(n^(4) - 1)` to be a multiple of 5, the last digit of
`(n^(4) - 1)` should be 0 or 5. In other words, the last digit of
`n^(4)` should be 1 or 6.

The last digit of
`n^(4)` is 1 if
`n` is an odd number except those that are multiples of 5.

The last digit of
`n^(4)` is 6 if
`n` is an even number except those that are multiples of 5.

Therefore, the possible values of
`n` are all positive integers except those that are multiples of 5.