A number theory problem by s p

Number Theory Level 3

Let $S$ be the set of all integers $n$ such that $\dfrac{8n^3-96n^2+360n-400}{2n-7}$ is an integer . Find the value of $\displaystyle \sum_{n\in S} |n|$

The answer is 50.

1 solution

Andrey Pershin
Aug 9, 2018

Remainder of this fraction is 27, so if we need to get integer number, the denominator of fraction $\frac{27}{2n-7}$ must be equal $\pm1$ , $\pm3$ , $\pm9$ or $\pm27$ . Possible values of n is $n=-10{,}-1{,}2{,}3{,}4{,}5{,}8{,}17$ . As a result the sum of absolute values is $\sum_{n} \vert n \vert\=50$

A number theory problem by s p

The answer is 50.

1 solution

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