Explanation:
If n is "any integer", then 2n+1 is "any odd number."
The square of any odd number is then ...
(2n+1)² = 4n² +4n +1 = 4n(n+1) +1
Since n is any integer, one of n and n+1 will be an even integer, so the product 4n(n+1) will be divisible by 8.
Then the sum 4n(n+1) +1 is one more than a number divisible by 8, hence ...
the square of an odd number is 1 more than a multiple of 8.
