If m and n are different positive integers, then how many prime number

Question

If m and n are different positive integers, then how many prime numbers are in set {m, n, m + n}?

(1) mn is prime.
(2) m + n is even.

Kudos for a correct  solution .

ENGRTOMBA2018 · Answer

Given: m,n > 0  \in  integer

Per statement 1, mn = prime. Now this can only happen if one of m or n =1 and the other is equal to 2 or 3 or 5 or 7 ...

Thus if mn =1*2 ----> {1,2,3} 2 primes, but if m =1, n=3, ---> {1,3,4} 1 prime. Thus this statement is not sufficient.

Per statement 2, m+n = even ---> both m and n are odd (or both are even, neglecting this as all of the cases for both even will give 0 primes!). ----> if m = 9, n = 15, ---> {9,15,24} 0 primes but if m = 1, n = 3 ---> {1,3,4} 1 prime. Thus again this statement is not sufficient.

Combining, we get,

mn = prime and m+n = even (either both even or both odd). Both even are ruled out as with 2 even numbers you will never get mn = prime. Thus the only case posible is both odd and mn = prime.

Thus the possible sets are :

{1,3,4} 1 prime
{1,5,6} 1 prime etc. Thus for all cases we get 1 prime and thus both statements combined are sufficient. C is the correct answer.

balamoon · Answer

Solution -

Stmt1  - mn is prime --> Product of two integers can be prime only if one integer must be 1 and the other one is prime. 
Lets take, m=1 and n=2. The set can be {1, 2, 3} – 2 prime numbers. If m=1 and n=3, set can be {1, 3, 4} – 1 prime number. Not Sufficient.

Stmt2  - m + n is even --> Integers m and n both must be even or odd. 
Lets take, m=1 and n=3. The set can be {1, 3, 4} – 1 prime number. If m=3 and n=5, set can be {3, 5, 8} – 2 prime numbers. Not Sufficient.

1+2   --> From both statements, the set is {1, n, n + 1}, where n is prime and odd and n + 1 is even. So n + 1 is not prime. Therefore, the set contains exactly one prime number. Sufficient.

ANS C.

ashokk138 · Answer

Consider St 1:

mn is prime. Product of two no.s is prime => one no should be 1.

Cases: 1 * 2 = 2 => Set {1, 2, 3} => 2 Prime no.s 
 1 * 3 = 3 => Set {1, 3, 4} => 1 Prime no.s (Any prime no other than 2, will have only 1 prime no.s in this set)

Two cases. Hence not sufficient.

St 2: m + n is even.

e + e = e ; o + o = e

Cases: 2 + 2 = 4 => Set {2, 2, 4} => 0 Prime no.s
 1 + 3 = 4 => Set {1 ,3, 4} => 1 Prime no.s
 1 + 15 = 16 => Set {1, 15, 16} => 0 Prime no.s

Not Sufficient.

Combining the two statements, 
only one case is possible. 1 * 3 = 3; 1 + 3 = 4; 
Hence Sufficient. Option C

longfellow · Answer

Answer is C

a prime number has only two factors, the number itself and 1. so either of m or n=1

also as 2 is also a prime number, 2 is also a potential number

2. makes it clear that the numbers are odd as odd+odd=even or 1+odd=even which leaves out 2 (1+2=3)

so the answer is C.

vishwaprakash · Answer

If m and n are different positive integers, then how many prime numbers are in set {m, n, m + n}?

(1) mn is prime.
(2) m + n is even.

Given , m,n >0
Asked : Prime in set {m,n,m+n}

Statement 1 : mn is prime,

Product can be prime, if one of the number if Prime and another is 1. Lets say m = 1 and n is prime number.
So set can {1,2,3} -> Two prime number in set - 2 and 3
or set can {1,5,6} -> one prime number in set - 5.

So no definitive answer - hence insufficient to answer the question.

Statement 2 : m + n is even
Hence both m and N are odd.

So set can {1,3,4} -> one prime number in set - 3
or set can {3,5,8} -> two prime number in set - 3 and 5.

So no definitive answer - hence insufficient to answer the question.

Combining option 1 and option 2:
As per Option 1
set can {1,2,3} -> Two prime number in set - 2 and 3
or set can {1,5,6} -> one prime number in set - 5.

out of two consecutive number only one can be prime, Only exception is 2 and 3, both are consecutive and both are prime.

As per option 2 : m has to be odd, since n is always 1 and will be always odd. So m cannot be 2. Hence Set will always contain 1 prime number.

Hence Option C

TimeTraveller · Answer

(1)  (mn)  is prime - the only possible way for  (mn)  to be prime is when one of them equals 1 and other is a prime number. In this case the set  {m, n, (m+n)}  may contain either 1 or 2 primes, depending on whether  m  or  n  is 2. NOT SUFFICIENT.

(2)  m+n  is even - this does not say anything about the set  {m, n , m+n}  because it could be any arbitrary number such that their sum is even. NOT SUFFICIENT.

(1)&(2) - the only case when  (mn)  and  (m+n)  are prime are when one of them equals 1 and the other equals a prime number. Hence, if we assume  m = 1  and  n = 2 , the set  {m, n, m+n}  contains  {1, 2, 3} = 2  primes.

Ans (C).

Bunuel · Answer

800score Official Solution:

Statement (1) tells us that mn is prime. The product of two integers can be prime only if one integer is 1 and the other one is prime. Otherwise the product would be divisible by more factors than just itself and 1. We can plug in some small prime numbers to see that statement (1) by itself is NOT sufficient. The set can be {1, 2, 3} – 2 prime numbers, or it can be {1, 3, 4} – 1 prime number.

Statement (2) tells us that m + n is even. Therefore the integers are both odd or both even. Clearly, there are many options possible: {2, 4, 6} – 1 prime number, {4, 6, 8} – no prime numbers, {3, 5, 8} – 2 prime numbers. Therefore statement (2) by itself is NOT sufficient.

If we use the both statements together, the set is {1, n, n + 1}, where n is prime and odd. If n is odd, then n + 1 is even. So n + 1 is NOT prime. Therefore the set contains exactly one prime number. Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient.

The correct answer is C.

bumpbot · Answer

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If m and n are different positive integers, then how many prime number

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