There exist several positive integers x such that 1/(x^2 + 2x) is a terminating decimal. What is the second smallest such integer?

Guest Jul 28, 2021

#1**-5 **

For starters, only numbers that divide over 2 or 5 are terminating decimals.

It is not hard to see that 2 is much smaller than 5, so trying x=2 and x=4 and we get 4 is the second smallest such integer.

Hope this helps.

OrangeJuicy Jul 28, 2021

#2**+1 **

Hi Orange Juice,

"For starters, only numbers that divide over 2 or 5 are terminating decimals."

What about 4 or 8 or any power of 2 or any power of 5 or any power of 10 ..................

Ohhh .... do you mean the denominator must be a power of 2 or a power of 5 or a power of 10 .........

Mmmm not sure what you mean ..............

Melody
Jul 28, 2021

#4**-5 **

Just to clarify what I meant by that:

Fractions whose denominator does not include 5 and 2 as factors are non-terminating decimals unless the numerator can cancel the factor that is not 2 or 5.

And this chart would be great to check out too.

Note that the four colors are different classifications of the examples.

OrangeJuicy
Jul 30, 2021