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If \(243^x*463^y =n\), where x and y are positive integers, what is the units digit of n?
The units digit of \(243^x\) is the same as the units digit of \(3^x\) and similarly the units digit of \(463^y\) is the same as the units digit of \(3^y\), so the units digit of \(243^x*463^y\) equals to the units digit of \(3^x*3^y=3^{x+y}\). So, knowing the value of \(x+y\) is sufficient to determine the units digit of \(n\).
(1) \(x + y = 7\). Sufficient. (As cyclicity of units digit of \(3\) in integer power is \(4\), units digit of \(3^7\) would be the same as of units digit of \(3^3\) which is \(7\))
Re: If 243^x*463^y = n, where x and y are positive integers [#permalink]
10 Sep 2013, 23:57
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Re: If 243^x*463^y = n, where x and y are positive integers [#permalink]
03 Mar 2014, 04:18
I have a doubt. Cyclicity of unit digit of 3 is 4. Hence we know that every fourth power of 3 (3^4, 3^8, 3^12) will have the same unit digit, 1. Hence when option B says x = 4, knowing that x and y are positive integers, we know that xy will be a multiple of 4. Unit digit of 3^4k is always 1 isn't it? Shouldn't this be sufficient information?
Re: If 243^x*463^y = n, where x and y are positive integers [#permalink]
03 Mar 2014, 04:21
Expert's post
siriusblack1106 wrote:
I have a doubt. Cyclicity of unit digit of 3 is 4. Hence we know that every fourth power of 3 (3^4, 3^8, 3^12) will have the same unit digit, 1. Hence when option B says x = 4, knowing that x and y are positive integers, we know that xy will be a multiple of 4. Unit digit of 3^4k is always 1 isn't it? Shouldn't this be sufficient information?
Shouldn't the answer be D?
I think you are missing that \(3^x*3^y=3^{x+y}\), so the exponent is x+y not xy.
Re: If 243^x*463^y = n, where x and y are positive integers [#permalink]
03 Mar 2014, 04:28
Expert's post
siriusblack1106 wrote:
Yes! Can't believe I just made that mistake. Such mistakes are gonna cost me. :/
Yes, careless errors are the #1 cause of score drops on the GMAT! They cause you to miss easier questions, hurting your score a lot more than not know how to solve the harder ones. So, be more careful, before you submit your answer, double-check that it’s the answer to the proper question. _________________
Re: If 243^x*463^y = n, where x and y are positive integers [#permalink]
02 Nov 2014, 17:48
Bunuel wrote:
If \(243^x*463^y =n\), where x and y are positive integers, what is the units digit of n?
The units digit of \(243^x\) is the same as the units digit of \(3^x\) and similarly the units digit of \(463^y\) is the same as the units digit of \(3^y\), so the units digit of \(243^x*463^y\) equals to the units digit of \(3^x*3^y=3^{x+y}\). So, knowing the value of \(x+y\) is sufficient to determine the units digit of \(n\).
(1) \(x + y = 7\). Sufficient. (As cyclicity of units digit of \(3\) in integer power is \(4\), units digit of \(3^7\) would be the same as of units digit of \(3^3\) which is \(7\))
(2) \(x=4\). No info about \(y\). Not sufficient.
Answer: A.
Hope it helps.
Hi Bunuel,
But surely shouldn't it matter if 3 is raised to 1 and or 6? Meaning, if it's 3^3 + 3^4 = 7 + 1 = 8. But, if its 3^2+3^5 = 9 + 3 = 12, units of 2. Doesn't that yield insufficient?
Re: If 243^x*463^y = n, where x and y are positive integers [#permalink]
03 Nov 2014, 00:50
Expert's post
russ9 wrote:
Bunuel wrote:
If \(243^x*463^y =n\), where x and y are positive integers, what is the units digit of n?
The units digit of \(243^x\) is the same as the units digit of \(3^x\) and similarly the units digit of \(463^y\) is the same as the units digit of \(3^y\), so the units digit of \(243^x*463^y\) equals to the units digit of \(3^x*3^y=3^{x+y}\). So, knowing the value of \(x+y\) is sufficient to determine the units digit of \(n\).
(1) \(x + y = 7\). Sufficient. (As cyclicity of units digit of \(3\) in integer power is \(4\), units digit of \(3^7\) would be the same as of units digit of \(3^3\) which is \(7\))
(2) \(x=4\). No info about \(y\). Not sufficient.
Answer: A.
Hope it helps.
Hi Bunuel,
But surely shouldn't it matter if 3 is raised to 1 and or 6? Meaning, if it's 3^3 + 3^4 = 7 + 1 = 8. But, if its 3^2+3^5 = 9 + 3 = 12, units of 2. Doesn't that yield insufficient?
Are you sure you are reading the question correctly? It's 243^x*463^y, 243^x multiplied by 463^y not 243^x + 463^y... _________________
Re: If 243^x*463^y = n, where x and y are positive integers [#permalink]
05 Jan 2016, 11:15
Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________
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