# Project Euler analyzing and simplifying problem 33

These questions concern "Project Euler problem 33":

The fraction $\displaystyle \frac {49}{98}$ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that $\displaystyle \frac {49}{98} = \frac 48$, which is correct, is obtained by cancelling the $9$'s.

We shall consider fractions like, $\displaystyle \frac{30}{50} = \frac {3}{5}$, 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.

1. Is it safe to assume that all fractions are smaller than $\displaystyle 1\cdot \left(\frac kk\right)$ ? ps my mistake, i understood something else

1. Is it, too, safe to assume that cancelling will be made only on the second digit of the numerator and on the first digit of the denominator?
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On your first point, the question says (my emphasis)

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.

On your second point, it is only a safe assumption if you know the four answers or can prove the point. For example it should be obvious that you cannot cancel both first (nonzero) digits, almost as obvious that you cannot cancel both second (nonzero) digits, but perhaps slightly less obvious for the first digit of the numerator and the second digit of the denominator.

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Oh, about the first I thought thought that mean something else. – Bakudan May 27 '11 at 14:52

Apart from the fact that the problem explicitly says "less than one," note that if we do not make that restriction then the product of however many such fractions there are would be trivially $1$.

Many of the Project Euler problems require a combination of programming and mathematics. If you can rule out certain sorts of possibilities by mathematical thinking, that may simplify the writing of a suitable program.

By Project Euler standards, this is a quite "small" problem, and it can be readily solved without the use of a computer, or indeed a calculator. Since you have been told there are only $4$ possibilities, if you believe the problem-setters are telling the truth, once you have found $4$ examples you are essentially finished. But in this case there is no need to have faith in the correctness of the assertion, since you can verify everything for yourself.

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