Question

For the fraction , assume that p is a prime number and q is not a multiple of p. For which of the following values of q will the decimal equivalent of be a repeating decimal with at least 1 nonzero digit that repeats?

1) 200
2) 80
3) 65
4) 40
5) 25

Answer 1

The values of q that will result in a repeating decimal with at least 1 nonzero digit that repeats are 65 and 25.

Step-by-step explanation:

The decimal equivalent of a fraction is a repeating decimal if and only if the fraction can be written in the form p/q, where p and q are integers and q is not divisible by any prime number other than 2 or 5. From the given options, we need to identify the values of q that are not divisible by any primes other than 2 or 5.

1. 200 is divisible by 2 and 5, so it does not meet the criteria.
2. 80 is divisible by 2 and 5, so it does not meet the criteria.
3. 65 is not divisible by any primes other than 5, so it can be the value of q.
4. 40 is divisible by 2 and 5, so it does not meet the criteria.
5. 25 is not divisible by any primes other than 5, so it can be the value of q.

Therefore, the values of q that will result in a repeating decimal with at least 1 nonzero digit that repeats are 65 and 25.