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

Globokim8 2020-11-17 Answered
To prove:
The given statement, " If k is any odd integer and m is any even integer, then k2 + m2 is odd".
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Expert Answer

Aniqa O'Neill
Answered 2020-11-18 Author has 100 answers
Proof:
The proof is given as,
Let k=2p + 1 and m=2q, where p and q are integers.
k2 + m2=(2p + 1)2 + (2q)2
=4p2 + 4p + 1 + 4q2
=2 (2p2 + 2p) + 1 + 2 (2q2)
=2l + 1 + 2m
where, l=2p2 + 2p, m=2q2 are integers.
k2 + m2=2 (l + m) + 1
k2 + m2=2n + 1, which is odd by definition.
where, n=l + m is an integer.
Thus, k2 + m2 is odd.
Conclusion:
The statement, " If k is any odd integer and m is any even integer, then k2 + m2 is odd" is proved.
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