Question

use Euclid's division Lemma t0 Show
that the square of any positive integer
is of the form 5n,5n+1 or 5n+4 where
'n' is a whole numbers,​

Answer 1

Let x be any integer

Then,

Either x=5m or x=5m+1 or x=5m+2 or, x=5m + 3 or x=5m+4 for integer x. [Using division algorithm]

If x=5m

On squaring both side and we get,

x² = 25m² = 5 (5m²) = 5n where n = 5m²

If x=5m+1

On squaring both side and we get,

x² = (5m+1)²

= 25m² + 1 + 10m

= 5 (5m²+2m) +1 (where 5m² + 2m = n)

= 5n + 1

If x = 5m + 2

Then x²

= (5m + 2)²

= 25m² + 20m + 4

= 5 (5m² + 4m) + 4

= 5n + 4 [Taking n = 5m² + 4m]

If x = 5m + 3

Then x²

= (5m + 3)²

= 25m² + 30m + 9

= 5 (5m² + 6m + 1) + 4

= 5n + 4 [Taking n = 5m² + 6m + 1]

If x = 5m + 4

On squaring both side and we get,

x² = (5m + 4)²

= 25m² + 16 + 40m

= 5 (5m² + 8m + 3) + 1 (where 5m² + 8m + 3 = n)

= 5n + 1

Hence, in each cases x² is either of the form 5n or 5n + 1 for integer n.