## Resilience

### Problem 243

Published on Saturday, 2nd May 2009, 10:00 am; Solved by 5905; Difficulty rating: 35%

A positive fraction whose numerator is less than its denominator is called a proper fraction.

For any denominator, `d`, there will be `d`−1 proper fractions; for example, with `d` = 12:

^{1}/_{12} , ^{2}/_{12} , ^{3}/_{12} , ^{4}/_{12} , ^{5}/_{12} , ^{6}/_{12} , ^{7}/_{12} , ^{8}/_{12} , ^{9}/_{12} , ^{10}/_{12} , ^{11}/_{12} .

We shall call a fraction that cannot be cancelled down a *resilient fraction*.

Furthermore we shall define the *resilience* of a denominator, `R`(`d`), to be the ratio of its proper fractions that are resilient; for example, `R`(12) = ^{4}/_{11} .

In fact, `d` = 12 is the smallest denominator having a resilience `R`(`d`) < ^{4}/_{10} .

Find the smallest denominator `d`, having a resilience `R`(`d`) < ^{15499}/_{94744} .