If each of the two digits X and Y is distinct

Question

If each of the two digits X and Y is distinct, is the two digit integer XY prime?

(1) Each of the digits X and Y is the sum of 2 distinct single digit prime numbers.

(2) The sum of digits X and Y is 16.

Bunuel · Answer

You are right, answer must be B:

(2) The sum of digits X and Y is 16. There are only three such two-digit numbers possible: 79, 97 and 88, but since we are told that X and Y are distinct then 88 is out and we are left with two prime numbers - 79 and 97.

Aximili85 · Answer

Hi Bunuel/Ahmed, could you please explain again as I'm getting a definite "no" from (1) and a "yes" from (2), which is not possible of course:
(1)The option XY = 75 can  not  exist, note it says that X & Y are both the  sum  of distinct prime numbers, so X and Y must each be one of the following options -> 4 (1+3), 6 (1+5), 8 (1+7 OR 5+3). So that gives only even XY options which can not be prime.

Thus (1) is a definite "NO", so its either A or D
(2) X+Y=16, so as you mentioned its 88, 79, 97 - 88 goes out as its not distinct for X & Y, so YES as they're both prime.

Not sure what I'm doing wrong here. Would appreciate your clarification!

Bunuel · Answer

1 is not a prime number. Single digit primes are: 2, 3, 5, and 7. So, x and y could be: 2+3= 5 , 2+5= 7 , 2+7= 9 , 3+5= 8 --> xy could be: 57, 75, 58, 85, 59, 95, 78, 87, 79, 97, 89 and 98. Some numbers are prime (for example 79) and some are not (for example 75). For more check Number Theory chapter of Math Book: math-number-theory-88376.html Hope it helps.

nutshell · Answer

Hi Bunuel,

I have a question here: I agree that the 2nd statement leads to 79, 97 and 88. While the 1st statement states that each of the digits in X and Y need to be single digit prime numbers, can we definitively conclude, using the 2nd statement, that 79 or 97 is a prime number? Since 9 is not a prime number, can both statements contradict or am I missing something here?

Bunuel · Answer

(1) doesn't say that X and Y must be primes, it says that "each of the digits X and Y is  the sum  of 2 distinct single digit prime numbers". So, x and y could be: 2+3= 5 , 2+5= 7 , 2+7= 9 , or 3+5= 8 .

nutshell · Answer

Hi Bunuel, Thanks for the quick reply. I misunderstood the statement in the first read.

BrentGMATPrepNow · Answer

Target question :    Is the two-digit integer xy prime?

Given : Each of the two digits x and y is  distinct

Statement 1 : Each of the digits x and y is the sum of 2 distinct single digit prime numbers.  
Let's TEST some values.
There are several values of x and y that satisfy statement 1. Here are two: 
 Case a : x = 3 + 5 = 8 and y = 2 + 7 = 9. So, xy = 89. In this case, the answer to the target question is  YES, xy IS prime 
 Case b : x = 2 + 7 = 9 and y = 3 + 5 = 8. So, xy = 98. In this case, the answer to the target question is  NO, xy is NOT prime 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : The sum of digits x and y is 16 
Since x and y are DISTINCT, there are only 2 possible ways to get a sum of 16. Let's examine each possible case: 
 Case a : x = 7 and y = 9. So, xy = 79. In this case, the answer to the target question is  YES, xy IS prime 
 Case b : x = 9 and y = 7. So, xy = 97. In this case, the answer to the target question is  YES, xy IS prime 
Since each possible case yields the SAME answer to the target question, it MUST be the case that  xy IS prime 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

EMPOWERgmatRichC · Answer

Hi All,

We're told that each of the two DIGITS X and Y is distinct. We're asked if the two-digit integer XY is PRIME. This is a YES/NO question and can be solved by TESTing VALUES.

1) Each of the digits X and Y is the sum of 2 distinct single digit prime numbers.

The single-digit prime numbers are 2, 3, 5 and 7. This means that X and Y could each be any of the following 4 numbers (keep in mind, they must be DISTINCT though - meaning different). 
(2+3) = 5 
(2+5) = 7 
(2+7) = 9 
(3+5) = 8

IF.... 
XY = 75 --> 75 is evenly divisible by 5, so it is NOT prime and the answer to the question is NO. 
XY = 79 --> 79 IS prime (trying dividing by 2 through 9 and you'll see) and the answer to the question is YES. 
Fact 1 is INSUFFICIENT

2) The sum of digits X and Y is 16.

Since the sum of the digits is 16, the only possible combination of digits is 7 and 9 (remember, the digits have to be DISTINCT, so XY cannot be 88). 
XY = 79 --> 79 IS prime (trying dividing by 2 through 9 and you'll see) and the answer to the question is YES. 
XY = 97 --> 97 IS prime (trying dividing by 2 through 9 and you'll see) and the answer to the question is YES. 
Fact 2 is SUFFICIENT

Final Answer:  B

GMAT assassins aren't born, they're made, 
Rich

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If each of the two digits X and Y is distinct

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