To prove: The given statement, " If k is any odd integer and m is any even integer, then k^{2} + m^{2} is odd".

Question

To prove:
The given statement, " If k is any odd integer and m is any even integer, then

k^{2} + m^{2}

is odd".

Answer 1

Proof:
The proof is given as,

Let k = 2 p + 1 and m = 2 q, where p and q are integers .

k^{2} + m^{2} = {(2 p + 1)}^{2} + {(2 q)}^{2}

= 4 p^{2} + 4 p + 1 + 4 q^{2}

= 2 (2 p^{2} + 2 p) + 1 + 2 (2 q^{2})

= 2 l + 1 + 2 m

where, l = 2 p^{2} + 2 p, m = 2 q^{2} are integers.

k^{2} + m^{2} = 2 (l + m) + 1

k^{2} + m^{2} = 2 n + 1, which is odd by definition.

where, n = l + m is an integer.

Thus, k^{2} + m^{2} is odd.

Conclusion:
The statement, " If k is any odd integer and m is any even integer, then

k^{2} + m^{2}

is odd" is proved.

Expert Answer