2015 · Q38
Let p be a prime number other than 2 or 5. One would like to express the vulgar fraction 1/p in the form of a recurring decimal. Then the decimal will be
Elementary MathematicsNumber System
- Aa pure recurring decimal and its period necessarily will be (p - 1)
- Ba mixed recurring decimal and its period necessarily will be (p - 1)
- Ca pure recurring decimal and its period will factor of (p- 1)
- Da mixed recurring decimal and its period will be some factor of (p - 1)
Explanation
For prime p ≠ 2, 5, gcd(10, p) = 1, so 1/p is purely periodic; its period is the multiplicative order of 10 modulo p, which divides p − 1 by Fermat's theorem (option C).