What will be the Largest possible value?

Number Theory Level 3

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


The answer is 890.

S P by s p , 16, India

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1 solution

Naren Bhandari
May 25, 2018

n 3 + 100 = n 3 + 1 0 3 − 900 = ( n + 10 ) ( n 2 − 10 n + 100 ) − 900 n^3+100 = n^3+10^3 -900 = (n+10)(n^2-10n+100)-900 Dividing by n + 10 n+10 we get n 3 + 100 n + 10 = n 2 − 10 n + 100 − 900 n + 10 \dfrac{n^3+100}{n+10}= n^2-10n+100-\dfrac{900}{n+10} Hence the largest possible value is n + 10 = 900 ⟹ n = 890 n+10=900 \implies n =890

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