Problem 33

Digit Cancelling Fractions

The fraction $49 / 98$ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that $49 / 98 = 4 / 8$ , which is correct, is obtained by cancelling the $9$ s.

We shall consider fractions like, $30 / 50 = 3 / 5$ , to be trivial examples.

There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.

If the product of these four fractions is given in its lowest common terms, find the value of the denominator.

消去数字的分数

$49 / 98$ 是一个有趣的分数，因为缺乏经验的数学家可能在约简时错误地认为，等式 $49 / 98 = 4 / 8$ 之所以成立，是因为在分数线上下同时消去了 $9$ 的缘故。

显然，存在诸如 $30 / 50 = 3 / 5$ 这样的平凡解。

除此之外，在所有值小于 $1$ 且分子和分母都是两位数的分数中，恰好有四个非平凡解。

将这四个分数的乘积写成最简分数，并求此时分母的值。