A number theory problem by Swadesh Rath

find the maximum value of n for which - (n^3 + 100 )/(n + 10 ) is an integer


The answer is 890.

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2 solutions

Curtis Clement
Aug 3, 2015

I used practically used the same method as Maggie, but I included the substitution step. i.e. u = n + 10 \ u = n+10 . I made this substitution because it is easier to deal with the fraction if the denominator is u rather than n+10.

Maggie Miller
Aug 3, 2015

For n 10 n\neq10 , n 3 + 100 n + 10 = n 2 10 n + 100 900 n + 10 \frac{n^3+100}{n+10}=n^2-10n+100-\frac{900}{n+10} .

Therefore, n 3 + 100 n + 10 \frac{n^3+100}{n+10} is an integer if and only if 900 n + 10 \frac{900}{n+10} is an integer, and the largest such n n is 890 \boxed{890} .

Where is 900/n+10 coming from ??

Nick Newman - 5 years, 10 months ago

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