Show that the square of any positive integer cannot be the form of 5m+2 where "m" is a whole number
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Step-by-step explanation:
Let a be any positive integer
By Euclid's division lemma
a=bm+r
a=5m+r (when b=5)
So, r can be 0,1,2,3,4
Case 1:
∴a=5m (when r=0)
a
2
=25m
2
a
2
=5(5m
2
)=5q
where q=5m
2
Case 2:
when r=1
a=5m+1
a
2
=(5m+1)
2
=25m
2
+10m+1
a
2
=5(5m
2
+2m)+1
=5q+1 where q=5m
2
+2m
Case 3:
a=5m+2
a
2
=25m
2
+20m+4
a
2
=5(5m
2
+4m)+4
5q+4
where q=5m
2
+4m
Case 4:
a=5m+3
a
2
=25m
2
+30m+9
=25m
2
+30m+5+4
=5(5m
2
+6m+1)+4
=5q+4
where q=5m
2
+6m+1
Case 5:
a=5m+4
a
2
=25m
2
+40m+16=25m
2
+40m+15+1
=5(5m
2
+8m+3)+1
=5q+1 where q=5m
2
+8m+3
From these cases, we see that square of any positive no can't be of the form 5q+2,5q+3.
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