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

Hi Balamoon,

You have made a mistake in evaluating the first statement. The unit digit of x is 6 and y is a positive Integer and by the principle of cyclicity, the number with unit digit 6 will always have unit digit 6 for any positive Integer exponent of the number

Hence the first statement alone answers the question here.

I hope it clears your mistake part in above working.

Both x and y are positive integers => Both x and y are not zero.
Keeping this in mind, let us approach the question.

(1) x = 26
Here, the units digit is 6 and 6^(any number) will result in 6 as the units digit.
So, x^y => 6 as units digit => Divided by 10 will give 6 as the remainder. Sufficient.

(2) y^x = 1

Now here we can have two possibilities:
(i) 1^x = 1, where x can be any number. So, x^y = x^1 = x.
We do not get a single remainder when x is divided by 10 as it is dependent on value of x. Not Sufficient
(ii) y^0 = 1 => 0^y = 0.
Here x = 0 is not possible because both x and y are POSITIVE INTEGERS
So option (ii) is not possible, and per (i) y^x = 1 is Not Sufficient

Many people, including me, make a mistake by assuming value of variables to be 0 even when the question explicitly states "positive intergers". DO NOT do that.

Answer is A.


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MANHATTAN GMAT OFFICIAL SOLUTION:

(1) SUFFICIENT: The tendency is to deem statement (1) insufficient because we have no information about the value of y. But 26 has a units digit of 6, and remember that 6^(any positive integer) has a units digit of 6 (the pattern is a single-term repeat).

6^1 = 6
6^2 = 36
6^3 = 216
etc.

Thus, 26 raised to ANY positive integer power will also have a units digit of 6 and therefore a remainder of 6 when divided by 10.

(2) INSUFFICIENT: Given that y^x = 1, there are a few possible scenarios:
However, the question stem tells us that x and y are POSITIVE integers, so we eliminate the first and third scenarios.

The remaining scenario indicates that y = 1 and x = any positive integer. Without more information about x, we cannot determine the remainder when x^y is divided by 10.

Since statement (1) tells us the value of x and statement (2) indirectly tells us the value of y (y = 1), the temptation might be to combine the information to arrive at an answer of C. This is a common trap on difficult Data Sufficiency problems. It might seem that we need both statements, when one statement alone actually provides enough information.

The correct answer is A.

Given
X and Y both positive.

Statement 1 :
X = 26

power cycle of 6 is 6, hence x ^ Y will always have unit's digit as 6 irrespective of Y. Henc X^y/10 will always have remainder as 6.

Hence statement 1 is sufficient.

Statement 2 :
y^x = 1
Now X > 0, hence Y = 1.

if Y = 1 and X is anything, then X^Y can be anything
Say x = 10, X^y/10 will give reminder as Zero
say X = 15, X ^Y/10 will give reminder as Five.

Statement 2 is insufficient.

Hence option A

X and Y = pos int
x^y / 10 = ? REMAINDER

1) 26^1 = 26/20 -> R = 6
26^2 = 676 / 10 -> R=6
26^3 = A large number ending in 6 -> R=6

S.

2) y^x = 1

Y = 1, x could be anything.

I.

A.

Rule of cyclisity...
Unit digit is 6 so the power doesn't matter...n the reminder will remain 6.
So option a.

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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: See the video below for tips on rephrasing the target question

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 that 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 that 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

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