Question

prove that the square os any positive integer is of the form 5q or 5q + 1 or 5q + 4 for some integer q.


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Answer 1

Step-by-step explanation:

Let the positive integer be a and b = 5.

By Euclid's division algorithm,

a = 5m + r, where 0 ≤ r < 5.

where r = 0,1,2,3,4.

If r = 0, then a = 5m

If r = 1, then a = 5m + 1

If r = 2, then a = 5m + 2

If r = 3, then a = 5m + 3

If r = 4, then a = 5m + 4

(i)

(5m)² = 25m²

         = 5(5m²)

         = 5q, where q is some integer.

(ii)

(5m + 1)² = 25m² + 1 + 10m

              = 5(5m² + 10m) + 1

              = 5q + 1, where q is some integer

(iii)

(5m + 4)² = 25m² + 16 + 40m

               = 5(5m² + 8m + 3) + 1

               = 5q + 1,  where q is some integer.

Hence, The square of any positive integer is either of the form 5q,5q+1 or 5q + 4, where q is some integer.

Hope it helps!