Critic Ivor Smallbrain is watching the classic film 11999 An

Critic Ivor Smallbrain is watching the classic film 11.999... Angry Men. But he is bored, and starts wondering idly exactly which rational numbers m/n have decimal expressions ending in 0000. . . (i.e., repeating zeroes). He notices that this is the case if the denominator n is 2,4,5,8,10 or 16, and he wonders if there is a simple general rule that tells you which rationals have this property.

Help Ivor by proving that a rational m/n (in lowest terms) has a decimal expression ending in repeating zeroes, if and only if the denominator n is of the form 2^a 5^b, where a,b 0 and a,b are integers.

Solution

If n is of the form 2^a * 5^b, then m/n ends in repeating zeroes
m/n = m/(2^a 5^b)
say a>b
then
m/(2^a 5^b) = m*5^(a-b) / 10^a
so clearly that expression ends in repeating zero,
similar sort of thing can be done if b>a.