Question

prove that if a and b are both odd integers, then ab+1 is an even integer

Answer 1

$\huge{\underline{\underline{\sf{AnSWeR}}}}$

If a and b are both odd then

a = 2m + 1 and

b = 2n + 1 for some integers m and n.

Then their product ab = (2m + 1)(2n + 1)

= 4mn + 2m + 2n + 1

= 2(2mn + m + n) + 1

and 2mn + m + n is an integer.

So if a and b are both odd, their product ab is odd. It follows that if ab is even, either a or b (or both) must be even.

Answer 2

Answer:

$\bf\huge\ Question:-$

prove that if a and b are both odd integers, then ab+1 is an even integer

$\bf\huge\ Answer:-$

If a and b are both odd then

a = 2m + 1 and

b = 2n + 1 for some integers m and n.

Then their product ab = (2m + 1)(2n + 1)

= 4mn + 2m + 2n + 1

= 2(2mn + m + n) + 1

and 2mn + m + n is an integer.

So if a and b are both odd, their product ab is odd. It follows that if ab is even, either a or b (or both) must be even.