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# If each of the two digits X and Y is distinct

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If each of the two digits X and Y is distinct  [#permalink]

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28 Aug 2012, 05:56
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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.
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Re: If each of the two digits X and Y is distinct  [#permalink]

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28 Aug 2012, 06:15
2
ahmed2502 wrote:
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.

OA. C
OE:
Statement (1) means that X or Y can take on the values 5, 7, 8, 9 each of which can be made by adding two distinct single digit prime numbers. Now, plug in to evaluate the statement. If XY = 75, then the answer to the question is no. However, if XY = 59, then the answer to the question is yes. The correct answer must be B, C or E. Statement (2) is also insufficient. If XY = 88, the answer is no but if XY = 79, the answer is yes. Cross off B. If the statements are combined, then only two numbers, 79 and 97, satisfy both conditions and both are prime.

I choose B. It is given X and Y are different, so XY=88 in second case is not possible.

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.
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Re: If each of the two digits X and Y is distinct  [#permalink]

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31 Aug 2012, 07:23
Bunuel wrote:
OE:
Statement (1) means that X or Y can take on the values 5, 7, 8, 9 each of which can be made by adding two distinct single digit prime numbers. Now, plug in to evaluate the statement. If XY = 75, then the answer to the question is no. However, if XY = 59, then the answer to the question is yes. The correct answer must be B, C or E. Statement (2) is also insufficient. If XY = 88, the answer is no but if XY = 79, the answer is yes. Cross off B. If the statements are combined, then only two numbers, 79 and 97, satisfy both conditions and both are prime.

I choose B. It is given X and Y are different, so XY=88 in second case is not possible.[/spoiler]

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!
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Re: If each of the two digits X and Y is distinct  [#permalink]

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31 Aug 2012, 07:39
Aximili85 wrote:
Bunuel wrote:
OE:
Statement (1) means that X or Y can take on the values 5, 7, 8, 9 each of which can be made by adding two distinct single digit prime numbers. Now, plug in to evaluate the statement. If XY = 75, then the answer to the question is no. However, if XY = 59, then the answer to the question is yes. The correct answer must be B, C or E. Statement (2) is also insufficient. If XY = 88, the answer is no but if XY = 79, the answer is yes. Cross off B. If the statements are combined, then only two numbers, 79 and 97, satisfy both conditions and both are prime.

I choose B. It is given X and Y are different, so XY=88 in second case is not possible.[/spoiler]

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!

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.
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Re: If each of the two digits X and Y is distinct  [#permalink]

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03 Sep 2012, 01:45
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?
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Re: If each of the two digits X and Y is distinct  [#permalink]

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03 Sep 2012, 01:50
1
nutshell wrote:
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?

(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.
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Re: If each of the two digits X and Y is distinct  [#permalink]

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03 Sep 2012, 01:59
Hi Bunuel,

Thanks for the quick reply. I misunderstood the statement in the first read.
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Re: If each of the two digits X and Y is distinct  [#permalink]

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25 Aug 2018, 05:47
Top Contributor
1
ConnectTheDots wrote:
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.

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

Cheers,
Brent
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Re: If each of the two digits X and Y is distinct  [#permalink]

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28 Nov 2018, 17:27
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

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Re: If each of the two digits X and Y is distinct   [#permalink] 28 Nov 2018, 17:27
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