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If x and y are positive integers such that the product of x

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If x and y are positive integers such that the product of x [#permalink] New post 24 Nov 2012, 12:21
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If x and y are positive integers such that the product of x and y is prime, what is the units’ digit of 7^x + 9^y?

(1) 24 < y < 32
(2) x = 1
[Reveal] Spoiler: OA

Last edited by Bunuel on 25 Nov 2012, 04:51, edited 2 times in total.
Renamed the topic and edited the question.
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Re: Can anyone help with this? [#permalink] New post 24 Nov 2012, 12:40
It should be 7^x and not 7X. If we consider 7X,solution would be E whereas if it is 7^x,solution is A.
1 states that y is between 24 and 32.As we know,product xy is prime.This is only possible if X is 1.
Considering Y as 29 gives us 9^29 and thus that leads us to its units digit as 1.Now 7^x+9^y = 1.
Similarly,Y = 31 gives us the same outcome.Sufficient

2 just gives us value of X.Insufficient.

Thus,ans is A
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Re: Can anyone help with this? [#permalink] New post 24 Nov 2012, 21:34
saikiranx wrote:
If x and y are positive integers such that the product of x and y is prime, what is the units’ digit of 7x + 9y?

(1) 24 < y < 32
(2) x = 1

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.


I do not agree with the OA. Answer should be E.

From the question statement, we can see that one of the numbers is 1 and the other is a prime.

1)We get x=1, y can be 29 or 31. If it is 29, units digit of expression is 8. If it is 31, units digit of expression is 6. Insufficient.

2) Obviously insufficient. As illustrated from statement 1.

Answer is hence E.

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Re: If x and y are positive integers such that the product of x [#permalink] New post 25 Nov 2012, 05:06
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If x and y are positive integers such that the product of x and y is prime, what is the units’ digit of 7^x + 9^y?

Since x and y are positive integers, then in order the product of x and y to be prime, either of them must be 1 another must be a prime number.

(1) 24 < y < 32 --> y is not equal to 1, thus y must be a prime number and x must be equal to 1. Only primes between 24 and 32 are 29 and 31, so y is either 29 or 31. Now, the units digit of 9^odd is 9, thus the units’ digit of 7^1 + 9^odd is 7+9=6. Sufficient.

(2) x = 1 --> y can be ANY prime number. If x=1 and y=2, then the units’ digit of 7^x + 9^y is 8, but if x=1 and y is any other prime then the the units’ digit of 7^x + 9^y is 6. Not sufficient.

Answer: A.

Hope it's clear.

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Re: If x and y are positive integers such that the product of x [#permalink] New post 27 Jan 2014, 13:14
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Re: If x and y are positive integers such that the product of x [#permalink] New post 01 Feb 2014, 03:34
Since x*y is prime one of them must be 1 and other must be prime.

Stmt1: 28<y<32. Therefore x has to be 1. and y must be 29 or 31. Now 9 raised to odd power always has 9 in units digit. What is this final units digit? 7+9 = 6. SUFF.
Stmt2: x=1. y could be any prime under the sun. NOT SUFF.
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Re: If x and y are positive integers such that the product of x [#permalink] New post 28 May 2014, 04:46
First things first. Xy is a prime number can only be true when either x or y is a prime number and the other is equal yo 1. Now, Statement 1 tells us that y is a prime number in that range thus y (29,31) and x of course 1. Now When 9 is raised to an odd number the units digit is always 9 therefore Statement 1 is sufficient.

Statement 2 tells us that x=1, but Y could be any prime number so it is insufficient.

Therefore A is the correct answer

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If x and y are positive integers such that the product of x [#permalink] New post 03 Sep 2014, 23:34
First, remember that a prime has only 2 factors: 1 and itself. Thus, if x * y = prime, either x or y is 1 and the other is prime.
Now, look at the cyclicity of 7 and 9. 7^1 = 7, 7² = 49, 7³ = 343, 7^4 = 2401 --> next will be units digit of 7 again. Cyclicity is 4.
For 9 the rule is : If 9 is raised to an odd power, the units digit will be 9, if its raised to an even power the units digit will be 1.

(1) 24 < y < 32 --> y is not 1, hence X has to be 1. Units digit of 7^1 = 7. Now the only two primes in the given interval are 29 and 31, both of which are odd. Hence units digit of 9^y will be 9. Hence units digit of the sum is 0. SUFF.

(2) x = 1. This tells us again that 7^1 = 7 BUT y can now be 2 (which is the only EVEN prime) OR any other ODD prime. Thus the units digit of 9^y can either be 1 or 9. Insufficient.

Hope its all clear!
If x and y are positive integers such that the product of x   [#permalink] 03 Sep 2014, 23:34
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