The maximum possible number of digits in the repeating block of the decimal representation of 1/p, where p is a prime number, is p-1.
Understanding Repeating Decimals
When a fraction is converted to a decimal, it can either terminate (end after a finite number of digits) or repeat indefinitely. If the denominator of the simplified fraction contains prime factors other than 2 and 5, the decimal representation will repeat. The repeating part is called the repetend or repeating block.
Maximum Length of Repeating Block
For a prime number p, the maximum possible length of the repeating block in the decimal representation of 1/p is p-1. This maximum length is achieved when 10 is a primitive root modulo p.
Example:
According to the provided reference, the length of the repeating block of 1/17 is 17-1 = 16. This demonstrates that the repeating block can, in some cases, reach the maximum possible length.
- For p = 7, the repeating block of 1/7 is 142857, which has a length of 6 (7-1).
- For p = 17, the repeating block of 1/17 has length 16.
- For p = 19, the repeating block of 1/19 has length 18.
