When I square an odd number, like 3, 5, or 7, I always get an odd square. For example, when I square 3, you get 9, and when I square 5, you get 25. Now, we have to see what happens when I divide these odd squares by 8.
Every time I divide an odd square by 8, you are left with a remainder of 1. So, no matter which odd square I choose, whether it's 9 or 25 or any other odd square, I will always have 1 left over when I divide it by 8.
When I square an even number, like 2, 4, or 6, I get an even square. For instance, when I square 2, I get 4, and when I square 4, I get 16.
Whenever I divide an even square by 8, I either have a remainder of 0 or 4. So, if I divide 4, 16, or any other even square by 8, I will always have either 0 or 4 left over.
These numbers leave a remainder of 7 when I divide them by 8. I know that odd squares leave a remainder of 1 when divided by 8, and even squares leave a remainder of 0 or 4.
Since 7 is not equal to 1, 0, or 4, it means that a number of the form 8n + 7 cannot be expressed as a sum of three squares. The remainders don't match up.