Hi uptowngirl,
First you have to understand what 'a terminating decimal' is.
lets say 1/2=0.5 but 1/3=.3333333333....... or 1/6=.1666666666........but for our convenience we write 1/3 as0.3 & 1/6 as 0.167 but this series is never ending(remaining is never zero) unlike 0.5 where the division ends when the remaining is zero.
Now take your problem-i)when x=3 then 3/32=.09375 and after this remaining is zero.that means it is a terminating decimal.
ii)x=5 then x/32=0.15625

iii)x=8 then x/32=8/32=1/4=0.25 and remaining is zero!

so answer is all.
Hope you understood as I have used very simple example.

I hope you understand a terminating decimal.


numbers liable to give a non terminating decimal are 3,6,7,9,11,....

if i have to define it then I would say that a number in denominator would yield a non terminating decimal
if the (number *10)(or factors of 10) is the only way a zero can be in its unit digit than it will Most time result in non terminating decimal. Why i said most I will explain later.

now when we divide a number by another number than for continued division we do the following
suppose we have to evaluate 1/3

the procedure is
multiply and divide by 10
=10/(10*3)

keep aside the 10 in denominator=1/10*(10/3)
=1/10*(3+1/3)----I hope you can understand this.
Now when will this terminate
only when 3*x=y0 where y is 1 or 2(2 remainders we can get by divison by 3 are 1,2)....x is from 1 to 9

thus you can easily see nos like 3,7,9,11,13,17....should all necessarily lead to a non terminating decimal. however nos like 6,12,15 may or may not result (thats why i said MOST)and nos like 2,4,8,10,16,20,25 will always have terminating decimal

why 6,12,15 may or may not terminate depends on if the numerator is divisible by 3 independently or not.
6=3*2
12=3*4
15=3*5

similarly we can see for 14,18 etc

100/12 will not terminate(100 is not divisible by 3
where as 75/12 would certainly terminate=25/4=6.25

If you have understood what i have said above than in order to check if a denominator will yield a terminating/non terminating decimal than just check for all prime factors of the denominator. if numerator does not divide prime factor like 3,7,11 etc it means it will be non terminating decimal.

See if you solve the question asked yourself.

here :

https://www.beatthegmat.com/terminating- ... 35070.html

Ian Stewart wrote:A terminating decimal is a decimal that stops. 1/5 = 0.2, 1/100 = 0.01, and 1/8 = 0.125 are all terminating decimals. 1/3 = 0.3333.... and 2/9 = 0.2222.... are not terminating (they're called 'repeating' or 'recurring' decimals). To recognize whether a fraction represents a terminating decimal:

1. Reduce your fraction completely
2. Prime Factorize the denominator. If the only prime factors of the denominator are 2, 5, or both, then it terminates. If there are any prime divisors of the denominator different from 2 or 5, it does not.

So if x is an integer and you have a fraction x/32 = x/2^5, since the only prime divisor of the denominator is 2, the decimal will terminate. It doesn't matter what x is.

I've explained this in more detail in earlier posts, which you may be able to find with a search.