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Is m+n prime?

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Is m+n prime?  [#permalink]

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New post 28 Mar 2020, 18:50
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Is m+n prime?
1) The difference between any two distinct positive factors of m is odd.
2) The difference between any two distinct positive factors of n is even.

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Re: Is m+n prime?  [#permalink]

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New post 28 Mar 2020, 18:50

Official Solution




St1) If the difference between any two positive factors is odd means one of the factors is Even and other is Odd.

We know, that 1 is a factor at all the integers. thus, to get the difference between any two positive factors is odd, we need another factor which is even

Only for 2, the factors would be 2 and 1 => difference as 1 = odd
hence, m = 2

NOT Sufficient

St 2) As per statement 2, n can be any odd integer.
NOT Sufficient

Even after combing
m=2, n = 5: m+n = 7 Prime
m=2, n = 7: m+n = 9 Not Prime
NOT Sufficient

Answer E
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Re: Is m+n prime?  [#permalink]

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New post 28 Mar 2020, 19:10
Is m+n prime?
1) The difference between any two distinct positive factors of m is odd.
2) The difference between any two distinct positive factors of n is even.

i) If the difference between any two positive factors is odd means one of the factors is Even and other is Odd.

Thus m has to be even, though m can be only 2 since for any other even number than 2 will have some factors which will both be even and thus the difference would be even in such cases.

Only for 2 the factors would be 2 and 1 => difference as 1 = odd

Insufficent as we don't know about n

ii) n can be any odd or even number For eg. 21 => 3 and 7 factors => difference = 4
m+n = 21+2 = 23 = prime
n can be 4 => Factors 2 and 4 => difference = 2
m+n = 4+2 = 6 = non prime

Answer - E
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Re: Is m+n prime?  [#permalink]

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New post 28 Mar 2020, 21:59
Asked: Is m+n prime?

1) The difference between any two distinct positive factors of m is odd.
m = 2; Since difference between 2-1= 1 = odd
If m = odd ; m-1 = even
And if m=even; m-1 = odd but m-2=even
m = 2 is the only possibility
Since n is unknown
NOT SUFFICIENT

2) The difference between any two distinct positive factors of n is even.
If n = even ; n-1 = odd; NOT FEASIBLE
But if n =odd; n-1=even; odd number has odd factors; odd-odd =even
n may be any odd number
but m is unknown
NOT SUFFICIENT

(1) + (2)
1) The difference between any two distinct positive factors of m is odd.
m = 2; Since difference between 2-1= 1 = odd
If m = odd ; m-1 = even
And if m=even; m-1 = odd but m-2=even
m = 2 is the only possibility
2) The difference between any two distinct positive factors of n is even.
If n = even ; n-1 = odd; NOT FEASIBLE
But if n =odd; n-1=even; odd number has odd factors; odd-odd =even
n may be any odd number
m + n = 2 + odd = odd
If m=2 & n = 3 ; m+n=5 ; prime number
But if m=2 & n=7; m+n = 9; Not a prime number
NOT SUFFICIENT

IMO E
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Re: Is m+n prime?  [#permalink]

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New post 28 Mar 2020, 22:01
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Is m+n prime?
1) The difference between any two distinct positive factors of m is odd.
2) The difference between any two distinct positive factors of n is even.


Lets test some numbers -

2 --> factors are 1,2 --> differences between those factors is 1, odd
3 --> ... 1,3 --> ... even
4 --> ... 1,2,4 --> ... both even/odd
5 --> ... 1, 5 --> ... even
8 --> ... 1,2,4,8 --> ... both even/odd
9 --> ... 1, 3, 9 --> ... even
10 --> ... 1,2,5,10 --> even/odd
27 --> ... 1,3,9,27--> ... even

The ONLY number that meets statement (1) is 2. every other even number that could meet this requirement also has two as a factor, which doesn't meet the statement requirements. m = 2.

Statement (2) tells us that n could be any odd number, since it's factors will all be odd and the differences between those factors is even. n = odd.

So to rephrase the question with our known information... "is an odd number, plus two, prime?" 3+2=5= prime, but 13+2=5=not prime. Insufficient with both statements together --> answer is (e).
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Re: Is m+n prime?  [#permalink]

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New post 28 Mar 2020, 22:42
Is m+n prime?

1) The difference between any two distinct positive factors of m is odd.
If, m = 2 (2.1) , 2-1 =1,
m= 6 (1,2,3,6), 6-1 = 5/ 3-1 = 2, not possible.
Means only 2 satisfies the condition of m. But we don't know n.
If, n= 1 , then m+n = prime and if n=2, then m+n = not prime. So insufficient.

2) The difference between any two distinct positive factors of n is even.
If, n = 3 (3,1) , 3-1=2
n = 5 (5,1), 5-1=4
n = 9 (1,3,9), 9-1=8/9-3=6/3-1=2
n= 11 (1,11), 11-1 = 10
n = 15 (1,3,5,15)
If, m= 0, and n=3, then m+n = prime and if m=1 and n=3 then m+n = not prime. So insufficient.

1) + 2)
m= 2 and n = 3 , then m+n = 5 =prime
m=2, n=5, then m+n = 7 =prime
m=2, n=9, then m+n = 11 =prime
m =prime, n = 15, then m+n = 17 =prime.
Same pattern repeats. So sufficient.
Ans. C
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Re: Is m+n prime?  [#permalink]

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New post 29 Mar 2020, 04:55
we have 2 and any prime number from statemtnt 2 . so answer is E
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Re: Is m+n prime?  [#permalink]

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New post 29 Mar 2020, 06:06
(E) Cannot say
The difference between two distinct positive factors is a multiple of the number itself.
1 implies m is odd, since the difference is odd
2 implies n can be odd or even, since the difference is even then it can be that either n is even or the multiple part is even
Even/Odd+Odd can be anything and hence we cannot say

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Re: Is m+n prime?  [#permalink]

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New post 29 Mar 2020, 06:37
(1) Number is 2, Difference between positive factors 1 and 2 is odd. We do not know n. Insufficient

(2) Number can be any prime, 3, 5, 7, 11

(1) + (2)
2+3 = prime
2+7 = 9 = not a prime
Insufficient

Answer: E
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Re: Is m+n prime?   [#permalink] 29 Mar 2020, 06:37

Is m+n prime?

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