Question

Use Euclid's division lemma to show that the square of any integer is either of the form 5m, 5m+1 or 5m+4 where m is a whole number


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Answer 1

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

a. If x = 5m,

x2 = (5m)2

= 25m2

= 5(5m2) =

5n (where n = 5m2 )

b. If x = 5m+1,

x2 = (5m+1)2

= 25m2+10m+1

= 5(5m2+2m)+1

= 5n+1 (where n = 5m2+2m ) 7

c. If x = 5m+4,

x2 = (5m+4)2

= 25m2+40m+16

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

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

Thus in each of three cases x2is either of the form 5n or 5n+1 for integer n.

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