Question

If m and n are odd positive integers, then m^2 + n^2 is even, but not divisible by 4.justify.

Answer 1

Solution:-
Any odd positive integer is of the form 2q + 1, where q is some integer.
Suppose m = 2x + 1 and n = 2y + 1, where x and y are some integers.
m²+n² = (2x + 1)² + (2y + 1)²
4x²+1+4x+4y²+1+4y
4(x²+y²+x+y) + 2
m²+n² = 4p + 2, where p = x²+y²+x+y
4p and 2 are even numbers ⇒ 4p + 2 is an even number.
m²+n² is even number and leaves remainder 2 when divide by 4.
So, m²+n² is even but not divisible by 4
Hence proved.

Answer 2

Step-by-step explanation:

Since m and n are odd positive integers, so let m = 2q + 1 and n = 2p + 1 ,

•°• m² + n² = ( 2q + 1 )² + ( 2p + 1 )² .

= 4( q² + p² ) + 4( q + p ) + 2 .

= 4{( q² + p² + q + p )} + 2 .

= 4y + 2 , where y = q² + p² + q + p is an integer .

•°• q² + p² is even and leaves remainder 2, when divided by 4 that is not divisible by 4.

Hence, it is solved

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