Is m+n prime?

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

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).

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

we have 2 and any prime number from statemtnt 2 . so answer is E

(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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(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

isn't this one missing an option of m=-2? would also have 1 and as divisors?

It doesn't change the answer, but just saying.