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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 19 Sep 2010, 05:45
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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

-----
I get that just from the question stem itself, you know either x or y must be equal to 1, thus B cannot be the answer. I can't figure out how you can tell the units digit of 9 to the Y just from the statement one.

Help?
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Re: Primes & Powers [#permalink] New post 19 Sep 2010, 07:02
Atrain13gm wrote:
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

-----
I get that just from the question stem itself, you know either x or y must be equal to 1, thus B cannot be the answer. I can't figure out how you can tell the units digit of 9 to the Y just from the statement one.

Help?


Since we know xy is a prime, it implies as you rightly pointed that one of them is 1.

(1) This implies y is not 1, hence x must be 1. Also, we know y must be a prime since xy is a prime. So y can only be 29 or 31. Either case, y is odd. What you need to note is that the units digit of 9^{Odd number} is always 9.
So combining with x=1, I know the units digit of the number in question must be 7+9 or 6. Hence, sufficient

(2) Not sufficient, as I know nothing about y

So answer is (a)

Further explanation
units digits of 9^x
x=1 9
x=2 1
x=3 9
x=4 1
...
The pattern is easy to spot and imagine why it happens. As the last digit is always multiplied by 9, leaving either 1 or 9 itself
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Re: Primes & Powers [#permalink] New post 19 Sep 2010, 07:06
Hi!
I see that noone answer your question yet.
I won't discuss about (2) because you already gave the solution.

(1) We know that 24 < y < 32 hence x = 1 ! As a matter fact, if we want a product to be prime we need to multiply 1 by a prime number. As y is much greater than 1, x = 1. Besides, I just said that has to be a prime number. So y = 29 or 31!
Now let's see how it goes for the unit digit of th power of 9 :

9^0 = 1
9^1 = 9
9^2 = 81
9^3 = 729
9^4 = ...1

so 9^k if k is odd gives the unit digit as 9.... As y = 29 or 31 (odd numbers), (2) is SUFFICIENT

ANS: A.

Hope it's clear.
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Re: Primes & Powers [#permalink] New post 19 Sep 2010, 08:21
interestingly i think but for the fact that 2 is a prime and an even prime at that, we are not able to answer based on (2). so we know x=1, so y has to be prime. all primes are odds (but for 2), if not for that, would we not be able to say 7 + 9^odd = ends in 6
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Re: Primes & Powers [#permalink] New post 19 Sep 2010, 10:30
when xy is some prime number, either x = 1 or y =1.
1. 24<y<32 ------> x =1 , y = 29 or 31
when y =29----> 7^1 + 9 ^ 29 = 7 + any number ending in 9 ----> unit digit 6
when y = 31 -----> 7^1 + 9 ^ 31 = 7 + any number ending in 9 ----> unit digit 6
1) is sufficient
2) not sufficient
Ans is A
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Re: Primes & Powers [#permalink] New post 19 Sep 2010, 10:43
one more thing to add here-----
cyclicity of numbers is always good in GMAT to remember -
any number ending with 0,1,5,6 raised to any power will have same unit digit as the number itself, cyclicity is 1
any number ending with 2,3,7,8 will have cyclicity 0f 4
for example 3^(n+1) will have same unit as 3^(n+5) or 3^(n+9) or so on....
where n >= 0
3 -> 9 -> 27 -> 81 ----
243 -> 729 -> xxx7 -> xxx1 ---
any number ending with 4,9 will have cyclicity of 2
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Re: Primes & Powers [#permalink] New post 19 Sep 2010, 15:01
Atrain13gm wrote:
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

-----
I get that just from the question stem itself, you know either x or y must be equal to 1, thus B cannot be the answer. I can't figure out how you can tell the units digit of 9 to the Y just from the statement one.

Help?


My approach was:
1. Y can be either 29 or 31.
Now y^29 or y^31 will always end in 9 and watever the power of X is the units digit is going to be the same.
So A.

2. X=1 is not enough for a conclusion.
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Re: Primes & Powers [#permalink] New post 17 Oct 2010, 03:23
The good thing is that 9^y when y is a prime number greater than 2, since all the primes greater than 2 are odd, the digit will always be 9.

So in the statement 1 it doesn't matter whether y is 29 or 31, we just need to know that y is not 2. Any range that doesn't contain 2 will be good.
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Re: Primes & Powers [#permalink] New post 20 Oct 2010, 18:17
good one.... ans : A
Re: Primes & Powers   [#permalink] 20 Oct 2010, 18:17
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