Question

prove that the square of any positive integer is of the form 5q, 5q+1,or 5q+4

Answer 1

Let a be any positive integer. Then, it is of the form 3q, 3q + 1 or 3q + 2:

a = 3q, or a = 3q+1, or a = 3q+2.

a² = (3q)² = 9q².

9q² can be written as 3(3q²). Now 3(3q²) can be written as 3x, where x = 3q².

a² = (3q+1)² = 9q² + 6q + q = 3(3q² + 2q) + 1.

3(3q² + 2q) can be written as 3y, where y = 3q² + 2q.

a² = 3y+1

a² = (3q+2)²& = 9q + 12q + 4 = 9q² + 12q + 3 + 1 = 3(3q² + 4q + 1) + 1

3(3q² + 4q + 1) can be written as 3z, where z = 3q² + 4q + 1

a² = 3z+1

Thus, a² = 3x = 3m, where m = x, and a² = 3y+1 or a² = 3z+1 ==> a² = 3m+1, since 3y+1 and 3z+1 are in the form 3m+1.

Thus, the square of any positive integer is of the form 3m or 3m + 1 for some positive integer m.