use euclid's division lemma to show that the square of any positive integer is of the form 5n or 5n + 1 or 5n + 4, where n is a whole number.
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Answer:
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
2
=5(5m
2
)=5n where n=5m
2
If x=5m+1
On squaring both side and we get,
x
2
=(5m+1)
2
=25m
2
+1+10m
=5(5m
2
+2m)+1(where5m
2
+2m=n)
=5n+1
If x=5m+2
Then x
2
=(5m+2)
2
=25m
2
+20m+4
=5(5m
2
+4m)+4
=5n+4 [ Taking n=5m
2
+4m]
If x=5m+3
Then x
2
=(5m+3)
2
=25m
2
+30m+9
=5(5m
2
+6m+1)+4
=5n+4 [ Taking n=5m
2
+6m+1]
If x=5m+4
On squaring both side and we get,
x
2
=(5m+4)
2
=25m
2
+16+40m
=5(5m
2
+8m+3)+1(where5m
2
+8m+3=n)
=5n+1
Hence, In each cases x
2
is either of the of the form 5n or 5n+1 for
integer n..
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