Can you prove this theory? (decimal period of rational numbers)?

Rational numbers as decimal fractions: Whenever you divide a whole number by a number that is not just a multiple of 2 or 5, you will receive a decimal fraction that never ends. For certain numbers there clearly is a period: one digit for 3 or 9, two digits for 11, 33 and 99, three digits for e.g. 27 and 31,... show more Rational numbers as decimal fractions:

Whenever you divide a whole number by a number that is not just a multiple of 2 or 5, you will receive a decimal fraction that never ends.

For certain numbers there clearly is a period: one digit for 3 or 9, two digits for 11, 33 and 99, three digits for e.g. 27 and 31, six digits for 7 or 13, etc.

Characteristic of the number of digits in the period is that there is a number which only consists of the digit 9 which is divisible by the number you divide through.

I'd like to prove that there is a period for all rational numbers whose divisor isn't divisible by 2 or 5.

Based on this observation, I can rephrase the theory as:

For every whole number not divisible by 2 or 5 there exists a multiple with only the digit 9.


I'm also interested whether this observation is true for other numeric systems, e.g. hexadecimal.

For every whole number in a numeric system based on N not divisible by the prime numbers which N consists of there is a multiple
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