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If p and q are positive integers, and the remainder obtained
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24 Aug 2020, 09:54
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If p and q are positive integers, and the remainder obtained when p is divided by q is the same as the remainder obtained when q is divided by p, which of the following is a possible value of pq?
Re: If p and q are positive integers, and the remainder obtained
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24 Aug 2020, 10:18
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Carcass wrote:
If p and q are positive integers, and the remainder obtained when p is divided by q is the same as the remainder obtained when q is divided by p, which of the following is a possible value of pq?
(A) 62 (B) 55 (C) 42 (D) 36 (E) 24
The remainder obtained when p is divided by q is the same as the remainder obtained when q is divided by p This information is indirectly telling us that p = q To explain why, let's see what happens if p does NOT equal q If that's the case, then one value must be greater than the other value. Let's see what happens IF it were the case that p < q.
What is the remainder when p is divided by q? Since p < q, then p divided by q equals 0 with remainder p
IMPORTANT RULE: When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0
What is the remainder when q is divided by p? Based on the above rule, we know that the remainder must be a number such that 0 ≤ remainder < p
Hmmmmm. In our first calculation (p ÷ q), we found that the remainder = p In our second calculation (q ÷ p), we found that 0 ≤ remainder < p Since it's IMPOSSIBLE for the remainder to both EQUAL p and BE LESS THAN p, we can conclude that it's impossible for p to be less than q.
Using similar logic, we can see that it's also impossible for q to be less than p.
So, it MUST be the case that p = q So, pq = p² = the square of some integer
Check the answer choices . . . only D is the square of an integer.
Re: If p and q are positive integers, and the remainder obtained
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24 Aug 2020, 15:41
1
Carcass wrote:
If p and q are positive integers, and the remainder obtained when p is divided by q is the same as the remainder obtained when q is divided by p, which of the following is a possible value of pq?
(A) 62 (B) 55 (C) 42 (D) 36 (E) 24
For One-Answer-Multiple-Choice questions, we can take following steps. 1. Simplify conditions and the problem in the question first. 2. We can exclude impossible cases from the choices.
We can put \(p \le q\) without loss of generality.
Assume \(p < q\). If \(q\) has a remainder \(r\) when \(q\) is divided by \(p\), then \(q = p \cdot a + r\) for some integer \(a\) where \(0 \le r < p\). When \(p\) is divided by \(q\), we have \(p = q \cdot 0 + p\) since we have \(p<q\). Thus \(p\) has a remainder \(p\) when it is divided by \(q\). Then we have \(p = r\) from the condition that reminders are equal. However it doesn't make sense, since we have \(r < p\) when \(q\) is divided by \(p\). The assumption contradicts.
Hence, we have \(p = q\), which means \(p \cdot q\) is a square integer. We can exclude A, B, C and E since they are not square integers.
Re: If p and q are positive integers, and the remainder obtained
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04 Dec 2024, 22:25
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