Question

Let p be a fixed prime number. A rational number m/n is called p-integral if p does not divide the denominator n. Prove the following:

q∈Q is p-integral for all primes p ⟺ q∈Z

Answer 1

To prove that q is p-integral if and only if q is an integer, we need to show two things: 1) If q is an integer, then q is p-integral for all primes p. 2) If q is p-integral for all primes p, then q is an integer.

Step-by-step explanation:

To prove that q is p-integral if and only if q is an integer, we need to show two things:

1. If q is an integer, then q is p-integral for all primes p.
2. If q is p-integral for all primes p, then q is an integer.

Proof of claim 1:

If q is an integer, then we can say q = a, where a is an integer. Now, for any prime p, we can say a = pq + r, where q and r are integers.

Since p is a prime, it does not divide r (because r < p). Therefore, p does not divide the denominator r, and q is p-integral.

Proof of claim 2:

If q is p-integral for all primes p, then q can be expressed as q = m/n, where m and n are integers and p does not divide n.

If we assume q is not an integer, then it can be written as q = a/b, where a and b are integers and b ≠ 0.

Since q is p-integral for all primes p, p does not divide b for any prime p. But that means p does not divide b in the reduced fraction form as well. Therefore, q is an integer.