How high can you go?

Number Theory Level 3

What is the largest integer value of $n$ such that $\dfrac{n^3+500}{n+5}$ is an integer?

The answer is 370.

1 solution

Calvin Lin Staff
May 13, 2014

We can rewrite the numerator as $n^3 + 500 = n^3 + 125 + 375 = (n+5)(n^2 - 5n + 25) + 375$ . Since $n^2 -5n + 25$ is an integer, so $n+5$ must divide $375$ for the fraction to be an integer. Thus, the largest value of $n$ is $n = 375 - 5 = 370$ .

How high can you go?

The answer is 370.

1 solution

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