Question

Prove that the square of any positive integer is of the form 5m or,5m+1 or,5m+4 for some integer m

Answer 1

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Answer 2

Answer:

Step-by-step explanation:

Let x be any positive integer

Then x = 5q or x = 5q+1 or x = 5q+4  for integer x.

If x = 5q, x2 = (5q)2 = 25q2 = 5(5q2) = 5n (where n = 5q2 )

If x = 5q+1, x2 = (5q+1)2 = 25q2+10q+1 = 5(5q2+2q)+1 = 5n+1 (where n = 5q2+2q )

If x = 5q+4, x2 = (5q+4)2 = 25q2+40q+16 = 5(5q2 + 8q + 3)+ 1 = 5n+1 (where n = 5q2+8q+3 )

∴in each of three cases x2 is either of the form 5q or 5q+1 or 5q+4 and for integer q.