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If $p$ is a positive prime number and $q$ is a multiple of $p$, which
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17 Oct 2025, 01:56
17 Oct 2025, 01:56
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If $p$ is a positive prime number and $q$ is a multiple of $p$, which of the following must be multiples of both $p$ and $q$ ?
Indicate all such numbers.
(A) \(p\)
(B) \(q\)
(C) \(p^2\)
(D) \(q^2\)
Indicate all such numbers.
(A) \(p\)
(B) \(q\)
(C) \(p^2\)
(D) \(q^2\)
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Re: If $p$ is a positive prime number and $q$ is a multiple of $p$, which
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01 Nov 2025, 10:04
01 Nov 2025, 10:04
Expert Reply
Given:
- $p$ is a positive prime number.
- $q$ is a multiple of $p$ (so $q=k \times p$ for some integer $k$ ).
We want to find which of the following are multiples of both $p$ and $q$ :
(A) $p$
(B) $q$
(C) $\(p^2\)$
(D) $\(q^2\)$
Explanation:
- A number that is a multiple of both $p$ and $q$ must be divisible by both $p$ and $q$.
- $p$ is divisible by itself, but unless $k=1, p$ is generally not divisible by $q$. So (A) is not necessarily a multiple of both.
- $q$ is divisible by $p$ and obviously itself. So (B) must be a multiple of both.
- $\(p^2\)$ is divisible by $p$, but to be divisible by $\(q=k p, p^2\)$ must be divisible by $k p$, i.e., $p$ divisible by $k$, which is not necessarily true unless $k=1$. So (C) is not necessarily a multiple of $q$.
- $\(q^2=(k p)^2=k^2 p^2\)$, is divisible by both $p$ and $q$. So (D) must be a multiple of both.
Final answer:
The numbers that must be multiples of both $p$ and $q$ are:
- (B) $q$
- (D) $\(q^2\)$
- $p$ is a positive prime number.
- $q$ is a multiple of $p$ (so $q=k \times p$ for some integer $k$ ).
We want to find which of the following are multiples of both $p$ and $q$ :
(A) $p$
(B) $q$
(C) $\(p^2\)$
(D) $\(q^2\)$
Explanation:
- A number that is a multiple of both $p$ and $q$ must be divisible by both $p$ and $q$.
- $p$ is divisible by itself, but unless $k=1, p$ is generally not divisible by $q$. So (A) is not necessarily a multiple of both.
- $q$ is divisible by $p$ and obviously itself. So (B) must be a multiple of both.
- $\(p^2\)$ is divisible by $p$, but to be divisible by $\(q=k p, p^2\)$ must be divisible by $k p$, i.e., $p$ divisible by $k$, which is not necessarily true unless $k=1$. So (C) is not necessarily a multiple of $q$.
- $\(q^2=(k p)^2=k^2 p^2\)$, is divisible by both $p$ and $q$. So (D) must be a multiple of both.
Final answer:
The numbers that must be multiples of both $p$ and $q$ are:
- (B) $q$
- (D) $\(q^2\)$
gmatclubot
Re: If $p$ is a positive prime number and $q$ is a multiple of $p$, which [#permalink]
01 Nov 2025, 10:04
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