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If x and y are positive integers, what is the remainder when x^y is
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12 Jun 2015, 03:20
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Re: If x and y are positive integers, what is the remainder when x^y is
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12 Jun 2015, 03:38
Bunuel wrote: 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
Kudos for a correct solution. CONCEPT: Remainder when a number is divided by 10 remains the unit digit of NumberQuestion Redefined: Find unit digit of x^y?Statement 1: x = 26Since y is a positive integer and the cyclicity of any number with unit digit 6 remain 6 always for any positive integer exponent of it Therefore x^y = 26^y will have Unit digit 6 for any positive Integer value of y Hence, SUFFICIENTStatement 2: y^x = 1i.e. y has unit digit 1 but since the value of x is unknown therefore unit digit of x^y can't be calculated Hence, NOT SUFFICIENTAnswer: option
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Re: If x and y are positive integers, what is the remainder when x^y is
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12 Jun 2015, 03:34
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 Stmt 1  y is not know. In Sufficient. Stmt 2  There are two possible solutions available. Lets take x=y=1, then y^x=1. Lets take x=0 and y=1, then y^x=1. Two possible solutions, In sufficient. 1+2 > x=26, then y must be 1 to validate 1^26=1. Remainder is 6. Sufficient. Ans C.
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Re: If x and y are positive integers, what is the remainder when x^y is
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12 Jun 2015, 03:42
balamoon wrote: 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
Stmt 1  y is not know. In Sufficient. Stmt 2  There are two possible solutions available. Lets take x=y=1, then y^x=1. Lets take x=0 and y=1, then y^x=1. Two possible solutions, In sufficient.
1+2 > x=26, then y must be 1 to validate 1^26=1. Remainder is 6. Sufficient.
Ans C. 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.
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Re: If x and y are positive integers, what is the remainder when x^y is
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13 Jun 2015, 03:15
Bunuel wrote: 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
Kudos for a correct solution. 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 SufficientMany 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.Press Kudos if you like it!



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Re: If x and y are positive integers, what is the remainder when x^y is
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15 Jun 2015, 06:05
Bunuel wrote: 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
Kudos for a correct solution. 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 singleterm 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: Attachment:
20150615_1659.png [ 27.94 KiB  Viewed 2315 times ]
However, the question stem tells us that x and y are POSITIVE integers, so we eliminate the first and third scenarios. Attachment:
20150615_1700.png [ 44.81 KiB  Viewed 2313 times ]
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.
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Re: If x and y are positive integers, what is the remainder when x^y is
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16 Jun 2015, 07:55
Bunuel wrote: 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
Kudos for a correct solution. 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



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Re: If x and y are positive integers, what is the remainder when x^y is
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17 Jun 2015, 02:03
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.



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Re: If x and y are positive integers, what is the remainder when x^y is
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07 Mar 2017, 10:50
Rule of cyclisity... Unit digit is 6 so the power doesn't matter...n the reminder will remain 6. So option a. Sent from my MotoG3 using GMAT Club Forum mobile app



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