Find the unit digit of 19^x + 27^y ?

Question

Find the unit digit of 19^x+27^y ? (1) The product of x and y is an odd prime number (2) x > y Source : https://www.GMATinsight.com

yashikaaggarwal · Answer

Statement 1: the product of x and y is odd number. So none of the two is 2. Hence there are many possible combinations for two. (Insufficient)

Statement 2: x>y,
X and y can be any no. (Insufficient)

Statement 1&2 together: no definite constraint given to determine values of x and y. (Insufficient)

IMO E

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rajatchopra1994 · Answer

Explanation:

Statement 1:  The product of x and y is an odd prime number 
Means either x or y has to be 1 other can be any odd prime number.
Example, let x=1, then y can be 3 or 5
also  x & y can be a fraction too 
So unit number will change for 19^x+27^y 
Not Sufficient

Statement 2: x > y
if x=3, y can be 1,2
unit number will change for 19^x+27^y 
Not Sufficient.

Combining 1 & 2,
Still not Sufficient
IMO-E

GMATinsight · Answer

Please check the Video explanation as mentioned below...

https://www.youtube.com/watch?v=W6DzUdv7GQ0

GMATinsight · Answer

yashikaaggarwal

I posted another question very similar to this.

https://gmatclub.com/forum/find-the-uni ... l#p2554430

You might want to answer that to ensure that your line of thinking was apt while solving this one.

yashikaaggarwal · Answer

I am searching that only. There are 5 questions with same statements, I am bit lost there.

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GMATinsight · Answer

The above post has link to that other question. There is very little difference. But the little differences are all that matter here in GMAT.

chetan2u · Answer

A few odd observations.
1) I doubt any question in actuals would ask for units digit in this way of some term with variables in power where variables could be fractions too. Although, one can learn that never suppose a variable to be an integer unless mentioned.
2) As far as your solution goes, it is correct as far as OA is concerned. But you have taken wrong examples.
a) x=3, and y =1......19^3+27^1 will be same as 9^3+7^1=9+7=16.
b) x=6, and y=1/2.....19^6+27^(1/2) will not be same as 9^6+7^(1/2). It will be  9^6+\sqrt{27} . now  \sqrt{27}  should be 5.xyz as it is between 5^2 and 6^2. 
Surprisingly the units digit will be 9^6+5.xyz = 1+5=6, same as the other example you took.
Thus, the examples actually point towards C as the answer. 
c) Of course x= 15 and y=1/3 would give units digit as 9+3 or 2, and answer would be E.

GMATinsight · Answer

Agreed chetan2u Thank you for checking that example!!!

I will change my example. Although we have not witnessed such question to find unit digit when number is decimal form but it's fundamentally apt question from GMAT standards.

Nipungupta9081 · Answer

IMO E

Statement 1
The product of X and Y is prime odd prime number. 
Which means Their product is not 2 , too many possibilities can arise.
Not Sufficient

Statement 2
X is greater then Y - which means if X is 5 , Y can be ,2,3,4 any number smaller then 5
Not Sufficient

Combining both the statements 
1 & 2 still doesn't get us anything concrete

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suchita2409 · Answer

This question emphasizes on why the fractions are so important!!

I chose C and I now know that I was wrong.Thank you chetan2u sir your posts are always very helpful.

AnirudhaS · Answer

(1) + (2) xy= ODD prime = any prime but 2
So let y=1, x=prime no. (since x>y)
with y=3 and y=5 we get unit digits 2(9+3=1 2 ) and 6(9+7=1 6 ) respectively
Insufficient

Answer:  E

Papist · Answer

We cannot verify x and y are both integers. Even assuming they are, there are still two solutions: 2 and 6. NOT SUFFICIENT.

ANSWER: E

D4kshGargas · Answer

I really can't understand why the answer is (E)

(1) x * y = Odd Prime 
Therefore either x = 1 or y = 1 and the other number is any prime number
Insufficient

(2) x > y 
Clearly insufficient

(1) + (2) 
x * y = odd prime number
x > y
Therefore y = 1
so units digit of 27^y = 7

And as we know cyclicity of 9
__9, __1, __9, __1, __9....

It produces a units digit of 9 when the power is odd..