# Find whether a given rational number has a terminating decimal expansion

Without actual division find whether the rational number $\dfrac{1323}{264600}$ is terminating or non terminating.

I know that to solve this, we have to convert the denominator into the formula $2^n 5^m$.

• If there are only powers of 2 and powers of 5 in the denominator then it will terminate – danimal Apr 26 '15 at 15:41

In order for you to be able to convert the denominator to a number of the form $2^n 5^m$, it needs to a number that is divisible by $2$ or $5$ and no other primes, and that is not the case for $264600$. But, as it turns out, you can make the denominator that way by "simplifying" the fraction: $$\frac{1323}{264600} = \frac{1}{200}.$$ Now observe that $200 = 2^3 5^2$.
$$264600 = (100)\cdot(2646) = 2^35^2\cdot1323$$