Question

Show that Square of any integer is of the form 5m ,5m+1 or 5m+4 for some integer m.
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Answer 1

By Euclid division Algorithm, we have

a = bq + r [ 0 ≤ r < b ] ........( 1 )

Let 'a' be a given integer.

Then,another number = b = 5

On putting b = 5 in ( 1 ) ,we get

a = 5q + r , [ 0 ≤ r < 5 => r = 0, 1,2,3,4 ]

If r = 0

So, a = 5q ⇒ a² = 25q² = 5 × ( 5q² ) = 5m [ where m = 5q² ]

If r = 1

So, a = 5q + 1 ⇒ a² = 25q² + 10q + 1 = 5 ( 5q² + 2q ) + 1 = 5m + 1 [ Where m = 5q² + 2q ]

If r = 2

So, a = 5q + 2 ⇒ a² = 25q² + 20q + 4 = 5 ( 5q² + 4q ) + 2 = 5m + 4 [ where m = 5q² + 4q ]

If r = 3

So, a = 5q + 3 ⇒ a² = 25q² + 30q + 9 = 5 ( 5q² + 6q ) + 9 = 5m + 9 [ where m = 5q² + 6q ]

If r = 4

So, a = 5q + 4 ⇒ a² = 25q² + 40q + 16 = 5 ( 5q² + 8q ) + 16 = 5m + 16 [ where m = 5q² + 8q ]

Therefore, Square of any integer is of the form 5m ,5m+1 or 5m+4 for some integer m.