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If x and y are positive integers, what is the unit digit of the

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If x and y are positive integers, what is the unit digit of the [#permalink] New post   25 Feb 2020, 22:38
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If x and y are positive integers, what is the unit digit of the expression \(6^{x+2} + y^6?​\)

(1) x=6​

(2) y=7​
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B
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Re: If x and y are positive integers, what is the unit digit of the [#permalink] New post   25 Feb 2020, 23:45
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kapil1 wrote:
If x and y are positive integers, what is the unit digit of the expression \(6^{x+2} + y^6?​\)

(1) x=6​

(2) y=7​



Question: what is the unit digit of the expression \(6^{x+2} + y^6?​\)

We know that cyclicity of Unit digit of
Number ending with {2, 3, 7, 8} = 4
Number ending with {0, 1, 5, 6} = 1
Number ending with {4, 9} = 2

So, we need only unit digit of y to answer the question because \(6^{x+2}\) will always have unit digit 6 irrespective of the value of x

Statement 1: x=6​

NOT SUFFICIENT

Statement 2: y=7

Unit digit of \(6^{x+2} + y^6?​\) = Unit digit of \(6+7^6\) = Unit digit of \(6+7^2\)= Unit digit of \(6+9\) = 5

SUFFICIENT

Answer: Option B
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Re: If x and y are positive integers, what is the unit digit of the [#permalink] New post   26 Feb 2020, 06:19
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kapil1 wrote:
If x and y are positive integers, what is the unit digit of the expression \(6^{x+2} + y^6?​\)

(1) x=6​

(2) y=7​


Solution


Step 1: Analyse Question Stem


    • x and y are positive integers.
    • We need to find the unit digit of expression \(6^{x+2} + y^6\)
      o Or, we need to find the value of (unit’s digit of \(6^{x+2} \)+ unit’s digit of \(y^6\))
         We know that cyclicity of powers of 6 is 1.
         So, irrespective of value of x, unit’s digit of \(6^{x+2} = 6\)
Thus, we only need to find the value of y.

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


Statement 1: \(x = 6\)
• We are not getting the value of y.
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.
Statement 2: \(y = 7\)
• Here we get the value of y ,i.e. \(y = 7\)
Hence, statement 2 is sufficient.
Thus, the correct answer is Option B
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If x and y are positive integers, what is the unit digit of the [#permalink] New post   28 Feb 2020, 10:58
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kapil1 wrote:
If x and y are positive integers, what is the unit digit of the expression \(6^{x+2} + y^6?​\)

(1) x=6​

(2) y=7​


Note a positive power of 6 will always have a unit digit of 6, this is because \(6 * 6 = 36\) so no matter how many 6's you multiply together you always end with a last digit of 6.
Hence we should pay attention to the \(y^6\) part, as \(6^{x+2}\) always has a unit digit of 6.

Statement 1: Don't know y, insufficient.
Statement 2: Gives y, sufficient.

Ans: B
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Re: If x and y are positive integers, what is the unit digit of the [#permalink] New post   29 Feb 2020, 12:03
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kapil1 wrote:
If x and y are positive integers, what is the unit digit of the expression \(6^{x+2} + y^6?​\)

(1) x=6​

(2) y=7​


Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

\(6^x\) has a unit digit \(6\) whatever \(x\) is.
Thus \(6^x+2 + y^6\) relies on the variable \(y\) only.

Thus, condition 2) is sufficient.

Therefore, B is the answer.
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Re: If x and y are positive integers, what is the unit digit of the [#permalink]
29 Feb 2020, 12:03
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