Integral divisors

Number Theory Level 5

Find the sum of all integers $n$ such that $\dfrac {n ^{2} +13 n +2} { 3n +5}$ is an integer.

1 solution

Mathh Mathh
Jun 14, 2014

$3n+5\mid n^2+13n+2\iff 3n+5\mid (3n+5)(\frac{1}{3}n+\frac{34}{9})-\frac{152}{9}$ $\iff 3n+5\mid (3n+5)(3n+34)-152\iff 3n+5\mid 152$

Note that one of the steps I used required $\gcd(3n+5, 9)=1$ , which is true for all $n\in\mathbb Z$ .

Since $3n+5\mid 152$ , we have the posibilities $-27$ , $-8$ , $-3$ , $-2$ , $-1$ , $1$ , $11$ , $49$ .

Add those up and the answer is: $\boxed{20}$ .

I didn't understand the 2nd step in which you omitted 9 from denominator

VIDIT LOHIA - 6 years, 8 months ago

I multiplied $(3n+5)(\frac{1}{3}n+\frac{34}{9})-\frac{152}{9}$ by $9$ .

mathh mathh - 5 years, 7 months ago

Y did u omit 9 from the denominator ?

Bala vidyadharan - 5 years, 7 months ago

I multiplied $(3n+5)(\frac{1}{3}n+\frac{34}{9})-\frac{152}{9}$ by $9$ .

mathh mathh - 5 years, 7 months ago