If x and y are positive integers, what is the remainder when 5^x is

Question

If x and y are positive integers, what is the remainder when 5^x is divided by y?

(1) x is an even integer.
(2) y = 3.

Kudos for a correct  solution .

GMATinsight · Answer

Question : what is the remainder when 5^x is divided by y?

Statement 1: x is an even integer

No information about y

Hence,   NOT SUFFICIENT

Statement 2: y = 3

@x=1, 5^1 when divided by 3 gives remainder = 2
@x=2, 5^2 when divided by 3 gives remainder = 1

Hence,   NOT SUFFICIENT

Combining the two statements

Remainder   = Remainder   = Remainder

For x to be even (-1)^even = +1 hence Remainder = +1
e.g.
@x=2, 5^2 when divided by 3 gives remainder = 1
@x=4, 5^4 when divided by 3 gives remainder = 1

Answer: Option  C

balamoon · Answer

If x and y are positive integers, what is the remainder when 5^x is divided by y?

(1) x is an even integer.
(2) y = 3.

Kudos for a correct  solution .

Solution -

For x>=1, 5^x is 5, 25, 125, 625 ......

Stmt1  - If x is even integer, then 5^x is 25, 625 .... We do not know the value of y. In sufficient.

Stmt2  - For y=3, remainders will repeat in 2, 1, 2, 1 .....Remainder is varying. In Sufficient.

Stmt1+ Stmt2   -

x is even and y=3, the remainders are 1, 1, 1, ....... Sufficient.

Thanks

Kudos Please.

vishwaprakash · Answer

My Attempt :

Given X,Y > 0
Unit digit of 5^X will always be 5. So Remainder of (5^x)/y will depend on Y
Say for 5^2/5 will give remainder as 0, where as 5^2/3 will give remainder as 1

Statement 1 :
x is an even integer.
Say for 5^2/5 will give remainder as 0, where as 5^2/3 will give remainder as 1
Not sufficient

Statement 2:
Clearly sufficient to give the reminder value.
Example :
5^2 / 3 leaves remainder = 1
5^3/3 leaves remainder = 1
Statement 2 is sufficient to answer

Answer Option B

GMATinsight · Answer

Hi vishwprakash, You seem to have made a mistake here. Check the highlighted part 5^3/3 leaves remainder = 2 whereas you have taken it as 1 So this statement will NOT be Sufficient as it leads to inconsistent values of Remainders. I hope It clears you mistake!

adityadon · Answer

clearly any statement alone is not enough , as it gives info about only 1 variable at a time

so combining 1 and 2 , 
5^2 / 3 = 25/3 so reminder 1
5^4 /3 = 125/3 so reminder 1. and so on.

So both statement toghether are enough

Answer = C

bluesquare · Answer

5^x will be a multiple of 5, and so it will have a units digit of 5.

1: tells us nothing about y. Insufficient.
2: 5/3 = 1 R 2. 25 /3 = 8 R 2. So insufficient.

Together: you can do examples and see that 5^2=25 = 3*8+1, 5^4 = 625 = 208*3+1.
But algebraically:
5 = 3 + 2. so 5*5 = (3+2)(3+2) = 3*3+2*2*3+2*2=3*(3+2*2)+2*2. 2*2 = 4 so it is really 3*(3+2*2+1)+1. This can be simplified to 3*I+1, where I is some integer.
Because 5^even will be a multiple of 5^2, it is basically (3I+1)^some exponent. (3I+1)^2=(3I)^2+2*3*I+1^2, but again (3I)^2+2*3*I = 3*some integer. So the remainder will be 1 again.
Therefore, (3I+1)*(3I+1)=3I+1, so (3I+1)^any exponent will be 3I+1, so it will leave a remainder of 1. So sufficient.

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION: (1) INSUFFICIENT: 5^(even integer means) that 5^x = 25, 625, 15,625, etc. All of these numbers end in 25. If y = 5, then the remainder equals 0. If y = 4, then the remainder is 1. Therefore we cannot determine the answer just by knowing this pattern of x. (2) INSUFFICIENT: Let's test some different values for x: https://gmatclub.com/forum/download/file.php?id=27032 The pattern is clear: when 5 is raised to an odd power, the remainder is 1, but when 5 is raised to an even power, the remainder is 2. However, with only Statement 2, we don't know whether x is even or odd. Combining the two statements, we know the pattern for the remainder when 5^x is divided by 3, and we know which term in that pattern applies. When 5^(even integer) is divided by 3, the remainder is always 1. Proving this theoretically is not trivial, but we don't need to do a theoretical proof. The correct answer is (C): BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 2015-06-22_1705.png

bumpbot · Answer

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If x and y are positive integers, what is the remainder when 5^x is

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