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

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If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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Updated on: 04 Aug 2010, 22:19
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Difficulty:

75% (hard)

Question Stats:

57% (01:58) correct 43% (02:17) wrong based on 164 sessions

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

(1) Both m and n are prime numbers.
(2) p + 1 is even.

_________________

But there’s something in me that just keeps going on. I think it has something to do with tomorrow, that there is always one, and that everything can change when it comes.
http://aimingformba.blogspot.com

Originally posted by aiming4mba on 04 Aug 2010, 21:49.
Last edited by aiming4mba on 04 Aug 2010, 22:19, edited 1 time in total.
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Re: If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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04 Aug 2010, 22:35
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aiming4mba wrote:
If m and n are positive integers and mn = p + 1, is m + n = p ?
a. Both m and n are prime numbers.
b. p + 1 is even.

Given: $$mn=p+1$$. Question: is $$m+n=p$$?

(1) Both m and n are prime numbers. If $$m=n=2$$, then $$p=3$$ and the answer is NO, as $$m+n=2+2=4\neq{p=3}$$ but if $$m=2$$ and $$n=3$$ then $$p=5$$ and the answer is YES, as $$m+n=2+3=p=5$$. Not sufficient.

(2) p + 1 is even --> $$p=odd$$. No info about $$m$$ and $$n$$. Not sufficient.

(1)+(2) Examples from (1) are still valid (as in both examples $$p=odd$$), hence we still have two different answers. Not sufficient.

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Re: If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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03 Sep 2010, 03:48
Hi,
I dont know whether my approach is correct or wrong but I suspect anser is C.
From St1: i know they are prime numbers
and from St2: i get P as odd number -
mn = p + 1 from here, in order to p+1 to be even out of m and n one should be 2.
so i get m= (p+1)/2 (if n=2).
=> m+n =p => (p+1)/2 + 2 = p => defenitly not equal to P.

Please let me know if my approach was wrong.

Thanks
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Re: If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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03 Sep 2010, 04:32
1
sikalvag wrote:
Hi,
I dont know whether my approach is correct or wrong but I suspect anser is C.
From St1: i know they are prime numbers
and from St2: i get P as odd number -
mn = p + 1 from here, in order to p+1 to be even out of m and n one should be 2.
so i get m= (p+1)/2 (if n=2).
=> m+n =p => (p+1)/2 + 2 = p => defenitly not equal to P.

Please let me know if my approach was wrong.

Thanks

OA is given in the first post, under the spoiler and it's E.

In my post above there are 2 cases given satisfying the stem and both statements and giving different answers to the question, thus proving that answer is E:
If $$m=n=2$$, then $$p=3=odd$$ and the answer is NO, as $$m+n=2+2=4\neq{p=3}$$;
If $$m=2$$ and $$n=3$$ then $$p=5=odd$$ and the answer is YES, as $$m+n=2+3=p=5$$.

Also why "(p+1)/2 + 2 = p => defenitely not equal to P" (the red part)? If you solve it for $$p$$ you'll get $$p=5$$ so $$n=2$$ and $$m=3$$.

Hope it helps.
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Re: If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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02 Feb 2011, 18:24
If m and n are positive integers and mn = p + 1, is m + n = p ?

S1: Both m and n are prime numbers.
S2: p + 1 and m are both even

A. S1 sf
B S2 sf
C both A and B together sf
D. Each sf
E. Neither sf nor together sf
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Re: If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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02 Feb 2011, 18:39
Test 1: M(2), N(3) P(5)

Meets criteria

Test 2: M(2), N(7) P(13)

Fails criteria

All criteria fail to meet test 2

Therefore (E)
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Re: If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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02 Feb 2011, 19:39
Let us analyze what the question is asking prior to looking at the statements given. We know that:

$$mn = p + 1$$

$$m + n = p?$$

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.

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Re: If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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03 Feb 2011, 03:25
It took me between 3 and 4 minutes to think and answer this: Solved it using numbers eventually.

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

Q: m + n = p ?

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

mn = p + 1

So, p is one less than mn

1. Started with lowest prime numbers
m=2, n=2 -> mn = 4, p=3: m + n = 4; 4<>3. Ans: No
m=2, n=3 -> mn= 6, p=5: m + n = 5; 5=5. Ans: Yes
Not sufficient.

2. p + 1 is even
p is odd.

Used the same data set and disapproved:

m=2, n=2 -> mn = 4, p=3(odd): m + n = 4; 4<>3. Ans: No
m=2, n=3 -> mn= 6, p=5(odd): m + n = 5; 5=5. Ans: Yes
Not sufficient.

Together:
Same data set. Not sufficient.

Ans: E
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Re: If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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05 Jan 2014, 12:40
aiming4mba wrote:
If m and n are positive integers and mn = p + 1, is m + n = p ?
a. Both m and n are prime numbers.
b. p + 1 is even.

This boils down to

Is m+n = mn-1

Statement 1

m,n are prime numbers
Let's number pick.

Mind you. if m and n are 2 and 3 then yes
If m and 3 are 2 and 5 then no

Insuff

Statement 2

p+1 is even, then p is odd

We get is mn even?

Both together

mn could be even as well as odd depending on whether the number 2 is included as one of both

Cheers!
J
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Posts: 359
Re: If m and n are positive integers and mn = p + 1, is m + n = p ?  [#permalink]

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12 Oct 2017, 13:30
If m and n are positive integers and mn = p + 1, is m + n = p ?
a. Both m and n are prime numbers.
m =2, n = 3; m = 3, n = 5...only (2)(3) = 5 + 1 suffices. (3)(5) = 14 +1; (2)(5) = 9 +1. Thus except, case 1 - but rest don't add up. So NOT SUFF.
b. p + 1 is even. No clue about m,n. Not Suff.

St 1 and 2: p +1 = even; m and n = both prime. Again only one case (2)(3) = 5 +1 works; rest don't - (2)(5) = 9 +1; (2)(13)= 26 (25 +1). So not sufficient. Ans E.
Re: If m and n are positive integers and mn = p + 1, is m + n = p ? &nbs [#permalink] 12 Oct 2017, 13:30
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