Re: If m and n are positive integers and mn = p + 1, is m + n = p ?
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02 Feb 2011, 20:39
Let us analyze what the question is asking prior to looking at the statements given. We know that:
\(mn = p + 1\)
We are asked does:
\(m + n = p?\)
Using what we know, we can rearrange this question as follows:
\(m + n = p?\)
\(m + n = mn - 1?\)
\(mn - m = n + 1?\)
\(m(n-1) = (n + 1)?\)
\(m = \frac{n+1}{n-1}?\)
Since we know that m and n are both positive integers, n can not be greater than 3, otherwise m will result in a value between 1 and 2. We also n can not be 1. Therefore, this leaves two distinct possibilities:
\((m,n) = (2,3),(3,2)\)
Now let's move on to solving the question knowing these conditions.
Statement 1: Both m and n are prime numbers.
2 and 3 are both prime numbers, but so are 11 and 17. We need to know specifically that m and n are 2 and 3.
Therefore, not sufficient.
Statement 2: p + 1 and m are both even.
All this really tells us is that m is even. Given the initial condition that mn = p + 1, if either m or n are given to be even, it follows that p + 1 must be even as well. Hence, the distinct subset of (2,3) still exists, as well as various other possibilities of an even number and any other number.
Therefore, not sufficient.
Both Statements Together
We know that m and n are prime numbers, and that m is even. So m must be 2. Unfortunately, n is only defined to be a prime number. This could be 3 (in which case the statement is satisfied), but it could be any other prime number as well.
Therefore, not sufficient.
Answer: E