Question

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​

Answer 1

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