If a and b are odd positive integers where a greater than b prove that a-b2 always a positive integer-

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Mathematics

Euclid's Division Lemma

If a and b ar...

Question

If a and b are odd positive integers where a greater than b prove that $\frac{a + b}{2}$ always a positive integer.

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Solution

Since $a$ and $b$ are $2$ positive integers
They can be expressed as $a = 2 n + 1; b = 2 m + 1$ where $n, m$ are integers
So the expression $\frac{a + b}{2}$ takes the form $\frac{2 m + 1 + 2 n + 1}{2} \frac{2 m + 2 n + 2}{2} m + n + 1$
Since $m . n$ are integers $m + n + 1$ is also a integer

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