The 1 percent

Number Theory Level 3

What is the smallest positive integer $n > 1$ for which fewer than $1 \%$ of the positive integers less than $n$ are factors of $n$ ?

As an explicit example, there are $5$ positive integers less than $6$ , and $3$ of these are factors of $6$ ; so $60 \%$ of the positive integers less than $6$ are factors of $6$ .

The answer is 103.

1 solution

Anweshan Bor
Nov 28, 2015

Any non-prime integer will have(excluding itself) its halve or just less as its largest divisor. so, for this problem max. possibility for a no. having <1% divisors among the positive integers less than it, the no. should be prime. Let the no. be n. So, its smallest divisor is 1, no less than n=n-1. So, [{1/(n-1)} x 100]<1 solving this inequality, n>101. the smallest prime no. after 101 is 103.(ans).

The 1 percent

The answer is 103.

1 solution

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