Lesson What is the length of the repeating part in the decimal form of the fraction 1/937 ?

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<H2>What is the length of the repeating part of the decimal form of the fraction 1/937?</H2>

<H3>Problem 1</H3>In the decimal form, the fraction 1/937 has a repeating part.
What is the length of this repeating part?

<B>Solution</B>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In this lesson, I will prove that the number 1/937 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;has a repeating sequence of decimal digits of length 936.

<pre>
Let 'a' be the ratio 1/937

a = {{{1/937}}}.

Notice that 937 is a prime number. It is neither of the form {{{2^k}}} nor {{{5^k}}}, where 'k' is an integer.

Therefore, from the general theory, 1/937, as a decimal, has the form

a = {{{1/937}}} = 0.RPPPPP...,

where R is some unique sequence of digits of length 'r,' and P is the repeating sequence of digits of length 'p'.

+------------------------------------------------+
    |   We consider here 'p' as the minimal length.  |
    +------------------------------------------------+

Therefore,

{{{10^r*a}}} is the decimal number of the form R.PPPPP...    (1)

while

{{{10^r*10^p*a}}} is the decimal number of the form RP.PPPPP...    (2)

Notice that the decimal parts of these two numbers, (1) and (2), are identical 0.PPPP...

Therefore, the difference between numbers (1) and (2) is an integer number N

{{{10^(r+p)*a}}} - {{{10^r*a}}} = {{{10^r*(10^p-1)*a}}} = N,    (3)

because taking the difference kills the identical decimal parts.

We can rewrite equation (3) in the form

{{{10^r*(10^p-1)*(1/937)}}} = N,

or, equivalently,

{{{10^r*(10^p-1)}}} = 937*N.    (4)

It tells us that the left side, {{{10^r*(10^p-1)}}}, is divisible by the prime number 937.

But since 10 is coprime to 937, we conclude from this that {{{10^p-1}}} is divisible by the prime number 937.

According to Fermat's Little Theorem, this implies that p = 936.

Thus, this proves that the repeating sequence of decimal digits of {{{1/937}}} has length 936.
</pre>

So, the statement formulated at the beginning of my post is proved.

As you see, the proof is elementary, but inexpressibly elegant.

It is a classic statement and a classic proof of number theory.

My other lessons in this site on miscellaneous problems on divisibility of integer numbers are

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