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655-705 (Hard)|   Exponents|      
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anilnandyala
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If 243^x*463^y = n, where x and y are positive integers 02 Oct 2010, 04:44
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55% (hard)

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59% (01:35) correct 41% (01:44) wrong based on 1668 sessions

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If \(243^x*463^y =n\) , where x and y are positive integers, what is the units digit of n?

(1) x + y = 7

(2) x = 4
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Bunuel
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If 243^x*463^y = n, where x and y are positive integers 02 Oct 2010, 04:56
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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 \(7\))

(2) \(x=4\). No info about \(y\). Not sufficient.

Answer: A.

Hope it helps.
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If 243^x*463^y = n, where x and y are positive integers 03 Mar 2014, 05:18
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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?
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If 243^x*463^y = n, where x and y are positive integers 03 Mar 2014, 05:21
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siriusblack1106
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?
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I think you are missing that \(3^x*3^y=3^{x+y}\), so the exponent is x+y not xy.

Does this make sense?
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If 243^x*463^y = n, where x and y are positive integers 03 Mar 2014, 05:26
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Yes! Can't believe I just made that mistake. Such mistakes are gonna cost me. :/
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Bunuel
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If 243^x*463^y = n, where x and y are positive integers 03 Mar 2014, 05:28
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siriusblack1106
Yes! Can't believe I just made that mistake. Such mistakes are gonna cost me. :/
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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.
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If 243^x*463^y = n, where x and y are positive integers 08 Jul 2020, 23:36
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If 243^x*463^y = n, where x and y are positive integers, what is the units digit of n?

(1) x + y = 7

(2) x = 4

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Solution


Step 1: Analyse Question Stem


    • \( 243^x*463^y = n\)
      o Where x and y are positive integers.
    • We need to find the unit digit of n.
      o Now, the unit digit of both \(243^x\) and \(463^y\) will follow the cyclicity of 3.
      o i.e. the unit digit of \(243^x\) = unit digit of \(3^x\)
      o And, the unit digit of \(463^y\) = unit digit of \(3^y\)
         Therefore, the unit digit of \( 243^x*463^y\) = The unit digit of \( 3^x*3^y\) = The unit digit of \(3^{x+y}\)
Thus, to find the unit digit of n we need to find the value of x + y.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE


Statement 1: x+y = 7
    • The unit digit of \( 243^x*463^y = n\) = The unit digit of \(3^{7}\)
    • We can easily find the unit digit of \(3^{7}\) and hence the unit digit of n from this statement.
Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E.

Statement 2: x = 4
    • From this statement we know the value of x but we don’t know y.
    • Therefore, we cannot find the unit digit of \(3^{x+y}\) and hence the unit digit of n from this statement.
Hence, statement 2 is NOT sufficient.
Thus, the correct answer is Option A.
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If 243^x*463^y = n, where x and y are positive integers 26 Oct 2024, 08:39
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