Prove the following claim If n is an integer between 0 and 1

Prove the following claim:

If n is an integer between 0 and 10, then n = a^2 +b^2 +c^2 +d^2 , for some integers a, b, c, d.

By using proposition.

Solution

The perfect squares that are less than 10 are :

0 , 1 , 4 and 9

Now, all we need to do is see if we can add 4 perfect square numbers and produce each of
0,1,2,3,4,5,6,7,8,9 and 10

0 = 0^2 + 0^2 + 0^2 + 0^2

1 = 1^2 + 0^2 + 0^2 + 0^2

2 = 1^2 + 1^2 + 0^2 + 0^2

3 = 1^2 + 1^2 + 1^2 + 0^2

4 = 1^2 + 1^2 +1^2 + 1^2

5 = 1^2 + 2^2 + 0^2 + 0^2

6 = 1^2 + 1^2 + 2^2 + 0^2

7 = 1^2 + 1^2 + 1^2 + 2^2

8 = 2^2 + 2^2 + 0^2 + 0^2

9 = 3^2 + 0^2 + 0^2 + 0^2

10 = 3^2 + 1^2 + 0^2 + 0^2

The above are just one among many possibilities for each number...

Since we have clearly shown here that each of 0,1,2,3,4,5,6,7,8,9,10 can be written of the form a^2 + b^2 + c^2 + d^2, the proposition has been proved