Why Are Repeating Decimals Less Than Denominator?

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Repeating decimals are always less than their denominator because the long division process generates a finite number of possible remainders. When dividing an integer by another integer, the remainders can only range from 0 to one less than the denominator, leading to a maximum of N-1 unique remainders. Once a remainder repeats, the division process starts to repeat, resulting in a repeating decimal. This concept can be illustrated through various examples of long division. Understanding this principle is crucial for grasping why the number of repeating digits is consistently less than the denominator.
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If the repeating decimal is equal to a/b, why is the number of repeating digits less than the denominator b?
This was a question on my homework, and I could not find mathematical proof for the conclusion that number repeating digits is always less than b.
 
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The key lies in doing long division (at least that's how I discovered it way back when); run through a few dozen examples, maybe you'll pick up on it.
 
I don’t wish to seem a bit impatient, but is it possible one could post the mathematical proof? We will most likely be going over this in class tomorrow, and I do not wish to be behind in the event I am not able to find the answer before then.
 
Okay - imagine performing long division of an integer by an integer. In each step you have a remainder. There are only N possible remainders (from 0 up to N-1). If one of them is 0 then you're done. Otherwise, of the other possible remainders (from 1 up to N-1) you can only go through N-1 steps before one of those very same remainders occurs once again - it's inevitable! At that point, each step repeats the previous ones and therefore your quotient must repeat over and over!
 
Thank you for your help, Tide and Hurkyl.
 

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