Problem 033

Project Euler

Digit canceling fractions

Problem 33

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 9s.

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.

For the canceling to work, one of the digit in the numerator must be the same as one of the digit in the denominator. Since only non-trivial examples are to be consider, no number (numerator or denominator) will contain any zero. Also, notice the following relationships:

\frac{cx}{yc} = \frac{x}{y} or \frac{xc}{cy} = \frac{x}{y}

where c, x and y denote a digit between 1 to 9, we have

y*cx=x*yc or y*xc = x*cy

With cx < yc, xc < cy and x < y in order for the fraction to be less than one. Therefore, looping through 1 to 9 for x, x+1 to 9 for y and 1 to 9 for c in this order, we can find the four solutions quickly.

Finally, the answer is \frac{\Pi(y_i)}{gcd(\Pi(x_i),\Pi(y_i))}