8. Prove that if x and y are both odd positive integers then x +y is even
but not divisible by 4.
De Tre Fuclid's algorithm to find the HCF of 1190 and 1445. Express the
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Answer:
We know that any odd positive integer is of the form 2q+1, where q is an integer.
So, let x=2m+1 and y=2n+1, for some integers m and n.
we have x
2
+y
2
x
2
+y
2
=(2m+1)
2
+(2n+1)
2
x
2
+y
2
=4m
2
+1+4m+4n
2
+1+4n=4m
2
+4n
2
+4m+4n+2
x
2
+y
2
=4(m
2
+n
2
)+4(m+n)+2=4{(m
2
+n
2
)+(m+n)}+2
x
2
+y2=4q+2, when q=(m
2
+n
2
)+(m+n)
x
2
+y
2
is even and leaves remainder 2 when divided by 4.
x
2
+y
2
is even but not divisible by 4.
may this help u
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