Question

The numbers a, p, and q are distinct positive integers, where p and q are prime. If a is divisible by both p and q, and b = a/(p*q), which of the following must be true?

A. b is not divisible by both p and q.
B. If b is divisible by r, then a is divisible by r.
C. b is a positive integer.

Answer 1

The answer to the question is that C. b is a positive integer must be true, as dividing a number by its prime factors results in a positive integer. The other options cannot be determined as true without additional information.

Step-by-step explanation:

The student is asking a question related to number theory and divisibility rules involving prime numbers. We have three distinct positive integers a, p, and q where p and q are prime, and a is divisible by both p and q. The value of b is given as a/(p*q).

Based on this information, we can confidently determine that C. b is a positive integer must be true. This is because when a positive integer that is divisible by two primes is divided by the product of those primes, the result is also a positive integer, assuming that the primes are factors of the integer.

Regarding options A and B:

• Option A: We cannot be sure without additional context because b could have more factors than just p and q.
• Option B: This is generally true but with the condition that r is not a multiple of either p or q. If r is a multiple of p or q, then a may not be divisible by r depending on the multiplicity of p and q in a.