Question

Use division algorithm to show that square of any positive integer is of the from 5m or 5m+1 or 5m+4 where m is a whole number?

Answer 1

Step-by-step explanation:

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,

x 2 =25m

=5 (5m)

=5n where n=5m

If x=5m+1

On squaring both side and we get,

x2 = (5m+1)2

=25m+1+10m

=5 (5m+2m)+1

= 5n + 1

where5m+2m=n

If x=5m+2

Then x2 = (5m+2)2

=25m+20m+4

=5(5m+4m)+4

=5n+4 [ Taking n=5m+4m]

If x=5m+3

Then x2 = (5m+3)2

=25m+30m+9

=5(5m+6m+1)+4

=5n+4 [ Taking n=5m+6m+1]

If x=5m+4

On squaring both side and we get,

x 2 = (5m+4)

=25m+16+40m

=5(5m+8m+3)+1

= 5n+1

where 5m+8m+3=n

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

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