DS - Prime

[karthikpandian19]

If m and n are positive integers and mn = p + 1, is m + n = p ?

Both m and n are prime numbers.
p + 1 and m are both even.

How to deal with this problem - odd&even method OR algebraic method?

[eagleeye]

Hi karthikpandian19:

The correct answer should be E. I am not sure whether odd/even or algebra would be faster but I did it using numbers for contradiction. Let me explain:

We are given mn=p+1, and asked whether m+n=p.

1. Both m and n are prime numbers.
If m, n are 2,3 then 2*3 = (2+3) + 1, (so m+n=p)
If m, n are 3,5 then 3*5 does not equal (3+5) + 1 (so m+n does not equal p)
INSUFFICIENT

2. p+1 and m are both even.
If m is even, then mn=p+1 must be even, using the same example as above where m=2, n=3, we know that
for that one, m+n=p, but if m=4 and n=3, then 4*3 is not equal to (4+3)+1, so insufficient.

At this point let's check them together. where m is an even prime and n is also a prime.
We have already checked m=2, n=3 and we know that in that case m+n=p
Let's check another one. Let m=2, n=5. In that case 2*5 is not equal to (2+5)+1.
Hence together they are insufficient as well. Hence E

Let me know if this helps

[GMATGuruNY]

If m and n are positive integers and mn = p + 1, is m + n = p ?

Both m and n are prime numbers.
p + 1 and m are both even.

How to deal with this problem - odd&even method OR algebraic method?

It is given that mn = p + 1.
Thus, if m+n = p, then mn = m+n + 1.
In other words, then the PRODUCT of m and n is one more than the SUM of m and n.

Question rephrased: Is the product of m and n one more than their sum?

Statement 1: m and n are prime.
Statement 2: p+1 (which is equal to the product of m and n) and m are even.
If m=2 and n=3, both statements are satisfied.
Here, the answer is YES: the product (2*3=6) is one more than the sum (2+3=5).
If m=2 and n=5, both statements are satisfied.
Here, the answer is NO: the product (2*5=10) is not one more than the sum (2+5=7).

Since the answer can be YES or NO, the two statements combined are INSUFFICIENT.

The correct answer is E.

[jcnasia]

I used algebra to solve this problem.

If m + n = p, then mn = (m+n) + 1
mn - m = n + 1
m(n - 1) = n + 1
m = (n + 1) / (n - 1)

At this point, I realized that n = 2 or n = 3. Otherwise, m will not be a positive integer.
So, (m,n) = (3,2) or (2,3)

Statement 1: Both m and n are prime numbers.
(m,n) = (2,3) satisfies the equation above.
(m,n) = (2,5) doesn't satisfy the equation.

Statement 2: p + 1 and m are both even.
(m,n) = (2,3) satisfies the equation above.
(m,n) = (2,5) doesn't.

Statement 1 & 2:
(m,n) = (2,3) satisfies the equation.
(m,n) = (2,5) doesn't.

[karthikpandian19]

OA is E