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Question

Use Euclid's division lemma to show that the square of any positive integer cannot be of the form 5m + 2 or 5m + 3 for some integer m.


Solution


Let a be any positive integer.
By Euclid's division lemma,
a = bm + r where b = 5
a = 5m + r
So, r can be any of 0, 1, 2, 3, 4
a = 5m + 1 when r = 0
a = 5m + 1 when r = 1
a = 5m + 2 when r = 2
a = 5m + 3 when r = 3
a = 5m + 4 when r = 4
So, 'a' is any positive integer in the form of 5m, 5m + 1, 5m + 2, 5m + 3, 5m + 4 for some integer m.
Case I: a = 5m
 a2=(5m)2=25m2 a2=5(5m2)
= 5q, where q=5m2
Case II: a = 5m + 1
 a2=(5m+1)2=25m2+10m+1 a2=5(5m2+2m)+1
= 5q + 1, where q=5m2+2m
Case III: a = 5m + 2
 a2=(5m+1)2=25m2+20m+4=25m2+20m+4=5(5m2+4m)+4
= 5q + 4 where q = 5m2 + 4m
Case IV: a = 5m + 3
a2=(5m+32)=25m2+30m+9=25m2+30m+5+4
= 5 (5m2 + 6m + 1) + 4
= 5q + 4 where q = 5m2 + 6m + 1
Case V: a = 5m + 4
 a2=(5m+4)2=25m2+40m+16
= 25m2 40m + 15 + 1
= 5(5m2 + 8m + 3) + 1
= 5q + 1 where q = 5m2 + 8m + 3
From all these cases, it is clear that square of any positive integer can not be of the form 5m + 2 or 5m + 3

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