Solving for the Largest Positive Integer n

[Trail_Builder]

Homework Statement

Find the largest positive integer n such that n^3 + 100 is divisible by n + 10.

Homework Equations

The Attempt at a Solution

The hint I've been given is to use (mod n + 10) to get rid of the n.

but i don't quite see how it would work :S

all my attempts have gotten nowhere, lol.

a little prod in the right direction would be nice :)

cheers

 

[gabbagabbahey]

My method would be to just use any polynomial dividing technique to find the remainder of (n^3+100)/(n+10). If n^3+100 is divisible by n+10, then the remainder will have to be an integer.

 

[Trail_Builder]

right, i think i may have a solution to this problem. can someone please check it for me :) thanks.


say that n^3 + 100 is divisible by n + 10.

then n^3 + 100 = 0 (mod n + 10)

n = -10 (mod n + 10)

so (-10)^3 + 100 = 0 (mod n + 10)

... 0 = 900 (mod n + 10)

we want to maximise n, and because the above line essentially means that 900 is an integer multiple of (n + 10), the maximum n would be when 900 = n +10

so n = 890...


is this reasoning correct?

thanks:)

 

[gabbagabbahey]

Looks good to me I got the same thing using polynomial division.