How can I find the smallest possible of full miles to get full kilometers?

Question

$1 \textrm{mile} = 1.609344 \textrm{km}$

I know that using $1000000$ miles I can move the decimal point and get a full number of $1609344 $km.

But how can I find the smallest amount of full miles that is equal to full kilometers and do the same for other conversions?

Answer 1

$$1.609344=\frac{25146}{15625}$$

So $15625$ miles is exactly $25146$ kilometers, and these are the smallest natural numbers because the above fraction is reduced.

Answer 2

Jared has a point. But actually, Wolfram alpha tells me that actually, 6336 kilometers is equal to 3937 miles, exactly.

Why? The original asker's value was not totally exact.

Answer 3

Keep eliminating the common factors from the numerator and denominator. It is obvious both are even by looking at their last digit, so lets start by dividing both numbers by 2.

If 1 mile$=$ 1.609344$\implies 1000000$miles$=1609344$.

Start with $\frac{1609344}{1000000}$. Divide both numbers by 2 and you get $\frac {804672} {500000}$. Divide both numbers by 2 again and you get $\frac{402336}{250000}$. Divide both numbers by 2 again, and repeat until the numbers can not be reduced by a common factor. Keep dividing like this and you will end up with$\frac{25146}{15625}$, which can not be reduced any further. (It should take you 6 divisions of both numbers by 2, which ultimately means that dividing both numbers by $2^{6}=64$ would have gotten your irreducible fraction in this case.)