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

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

Problem 1

Let 'a' be the ratio 1/937

    a = .


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

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

    a =  = 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, 

     is the decimal number of the form R.PPPPP...    (1)

while 

     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

     -  =  = N,    (3)

because taking the difference kills the identical decimal parts.


We can rewrite equation (3) in the form

     = N,

or, equivalently,

     = 937*N.    (4)


It tells us that the left side, , is divisible by the prime number 937.


But since 10 is coprime to 937, we conclude from this that  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  has length 936.