1) prove that the sum of two squared numbers isn't always odd

2) prove that the sum of two consecutive numbers is a multiple of four

3) prove that the sum of three consecutive even numbers is a multiple of six

4) prove that the sum of four consecutive odd numbers is a multiple of eight

1. Here's an example! 2 squared plus six squared is 40, an even number.
2. n+(n+1) is divisible by four? Well, that's actually not always true. 2+3 is five, and 6+7 is 13. In fact, no 2 consecutive numbers will add up to four, because n +(n+1) is equal to 2n +1 which will always be odd. Sorry!
3.(n+(n+1)+(n+2))x2 (because even numbers are every other number) is divisible by six? This equals 6n+6, which will always be divisible by 6. 
4. 2n+1 (which is the equation for an odd number, try it!) +(2n+1+2)+(2n+1+4)+(2n+1+6) is divisible by 8? That equals 8n+16, which will always be divisible by eight. Hope this helps!!