Question 18

For $$n\ge 2$$ and $$n\in\mathbb{Z}$$, the smallest positive integer $$n$$ for which none of the fractions $$\frac{17}{n+17},\frac{18}{n+18},\frac{19}{n+19},\ldots,\frac{100}{n+100}$$ can be simplified is ______.

Correct Answer: 101

Solution

For each numerator $$k$$, $$\gcd(k,n+k)=\gcd(k,n)$$. Therefore $$n$$ must be coprime to every integer from $$17$$ through $$100$$. Every prime at most $$100$$ divides at least one number in this range, so the smallest possible prime factor of $$n$$ is $$101$$, and the least valid value is $$101$$.

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