Question

N is a whole number.

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Prove that (n - 10)(4n - 10) - 3 is always an odd number.
Input note: write your expression in the form 2(...) + 1 to clearly show this is an odd
number for all integer values of n.

Answer 1

`(n-10)(4n-10) -3\\\\=4n^2 -10n-40n+100-3\\\\=4n^2-50n+97\\\\=4n^2 -50n+96 +1\\\\=2(2n^2-25n+48)+1\\\\2(2n^2 -25n+48)~ \text{is even, so}~ 2(2n^2 -25n+48) +1~ \text{is odd.}`

`\text{Hence,}~ (n - 10)(4n - 10) - 3 ~ \text{is always an odd number for}~ n\in \mathbb{W}.`