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If x and y are positive integers, what is the remainder when

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by fskilnik@GMATH » Mon Oct 22, 2018 2:56 pm

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BTGmoderatorLU wrote:Source: Manhattan Prep

If x and y are positive integers, what is the remainder when x^y is divided by 10?

1. x = 26
2. y^x = 1
$$x,y\,\,\, \ge 1\,\,\,{\rm{ints}}\,\,\left( * \right)$$
$$?\,\, = \,\,\left\langle {\,{x^{\,y}}} \right\rangle \,\,{\rm{ = }}\,\,{\rm{units}}\,\,{\rm{digit}}\,\,{\rm{of}}\,\,\,{x^{\,y}}$$
$$\left( 1 \right)\,\,x = 26\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,? = \left\langle {\,{{26}^{\,y}}} \right\rangle = 6\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\,\,\,\,\left[ {\,\left\langle {\,{6^{\,y}}} \right\rangle = 6\,\,,\,\,\,\forall \,\,y\,\, \ge 1\,\,{\mathop{\rm int}} \,} \right]\,\,\,\,\,\,\,\,\,$$
$$\left( 2 \right)\,\,{y^x} = 1\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,\,{\rm{ = }}\,\,\,\left\langle 1 \right\rangle \,\,\, = \,\,\,1 \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,\,{\rm{ = }}\,\,\,\left\langle 2 \right\rangle \,\,\, = \,\,\,2\,\, \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.\,$$

This solution follows the notations and rationale taught in the GMATH method.

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Fabio.
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by Brent@GMATPrepNow » Tue Oct 23, 2018 5:14 am

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BTGmoderatorLU wrote:Source: Manhattan Prep

If x and y are positive integers, what is the remainder when x^y is divided by 10?

1. x = 26
2. y^x = 1
Given: x and y are positive integers

Target question: What is the remainder when x^y is divided by 10?
This is a good candidate for rephrasing the target question.
Notice that 43 divided by 10 leaves remainder 3, and 127 divided by 10 leaves remainder 7, and 618 divided by 10 leaves remainder 8.
So, asking for the remainder when x^y is divided by 10 is the same as asking what the units digit of x^y is. So, .....
REPHRASED target question: What is the units digit of x^y ?

Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Statement 1: x = 26
IMPORTANT: if y is a positive integer, then 26^y will always have units digit 6.
Notice that 26^1 = 26, and 26^2 = 676, and 26^3 = ????6, 26^4 = ????6, etc.
So, the answer to the REPHRASED target question is the units digit of x^y is 6
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: y^x = 1
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 11 and y = 1 (notice tat y^x = 1^11 = 1. In this case, x^y = 11^1 = 11. So, the answer to the REPHRASED target question is the units digit of x^y is 1
Case b: x = 12 and y = 1 (notice tat y^x = 1^12 = 1. In this case, x^y = 12^1 = 12. So, the answer to the REPHRASED target question is the units digit of x^y is 2
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent
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by ceilidh.erickson » Tue Oct 23, 2018 9:24 am

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BTGmoderatorLU wrote:Source: Manhattan Prep

If x and y are positive integers, what is the remainder when x^y is divided by 10?

1. x = 26
2. y^x = 1

The OA is A
It's important to note here that certain digits will always maintain the same units digit, regardless of the exponent: 0, 1, 5, and 6. All other digits [2, 3, 4, 7, 8, and 9] will change units digits, depending on the exponent. More on establishing those patterns here:
https://www.beatthegmat.com/what-is-the ... tml#800962
https://www.beatthegmat.com/if-a-is-an- ... 03798.html

So, if statement 1 had said "x = 24" or "x = 27," then it would not have been sufficient. We would also have needed information about the exponent. It is only because 6 maintains the same units digit regardless of the exponent that (1) alone was sufficient.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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