What is the units digit of x² - y² ?

Question

What is the units digit of x2 - y2 ?

(1) x - y is a positive multiple of 3
(2) x + y is a positive multiple of 10

Archit3110 · Answer

target find 
units digit of x² - y²

#1
x - y is a positive multiple of 3
possible pairs of x,y
3,0 ; 6,3 ; 9,6 ; 9,3
we get different values to target
insufficient
#2
x + y is a positive multiple of 10
possible pairs of x , y
9,1 ; 8,2 ; 7,3 ; 6,4 , 5,5 ; 
or it could be 9.5 1.5
target value is always 0,5
insufficient
from 1 &2
both x,y are integers 8,2 
sufficient
option C

gmatophobia · Answer

Hi  BrentGMATPrepNow  - Request you share an OE to this question. I am getting C as the answer, however the published OA is E.

Sharing my working, can you let me know if there is a flaw in my understanding. Thanks in advance for your response !

(1) x - y is a positive multiple of 3

We don't have sufficient information on the nature of x and y. Also the unit digit of (x+y)(x-y) will depend on the unit digit of (x+y) hence this statement is not sufficient.

We can eliminate A and D.

(2) x + y is a positive multiple of 10

x and y can be fractions such that when added the result is a positive multiple of 10. However we cannot conclude on the nature of x-y. Hence this statement by itself is insufficient.

Eliminate B

Combination

x - y is a positive multiple of 3 , therefore x-y is an integer
x + y is a positive multiple of 10, therefore x+y is an integer

(x+y)(x-y) is an integer with unit digit as 0.

Hence C

BrentGMATPrepNow · Answer

Target question :    What is the units digit of x² - y² ?

Strategy: When I scan the two statements, I see that they're related to Integer Properties (divisors, multiples, etc.). So, the first question I ask myself is "Are the variables defined as  integers ? Since we're NOT told x and y are integers, we need to watch out for that.

Statement 1 : x - y is a positive multiple of 3  
There are several values of x and y that satisfy statement 1. Here are two: 
 Case a : x = 6 and y = 3. In this case, x² - y² = (x + y)(x - y) = (6 + 3)(6 - 3)² = (9)(3) = 2 7 , which means the answer to the target question is  the units digit of x² - y² is  7  
 Case b : x = 7 and y = 4. In this case, x² - y² = (x + y)(x - y) = (7 + 4)(7 - 4)² = (11)(3) = 3 3 , which means the answer to the target question is  the units digit of x² - y² is  3  
Since we can’t answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : x + y is a positive multiple of 10  
There are several values of x and y that satisfy statement 2. Here are two: 
 Case a : x = 6 and y = 4. In this case, x² - y² = (x + y)(x - y) = (6 + 4)(6 - 4)² = (10)(2) = 2 0 , which means the answer to the target question is  the units digit of x² - y² is  0  
 Case b : x = 9.75 and y = 0.25. In this case, x² - y² = (x + y)(x - y) = (9.75 + 0.25)(9.75 - 0.25)² = (10)(9.5) = 9 5 , which means the answer to the target question is  the units digit of x² - y² is  5

Statements 1 and 2 combined    
If x - y is a positive multiple of 3, then x - y is an INTEGER
Likewise, If x + y is a positive multiple of 10, then x + y is an INTEGER
So, we can be certain that x² - y² = (x + y)(x - y) = (some multiple of 3)(some multiple of 10) = some integer  with units digit  0 .  
Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

BrentGMATPrepNow · Answer

You're absolutely right. The correct answer is C. 
I have edited the OA accordingly.

Kudos for you!!!

deepak.brijendra · Answer

Hi 
x^2-y^2= (x-y)(x+y). So no matter what x-y is if X+Y is multiple of 10 then the unit digit has to be 0. Plz let me know if I am wrong.

bumpbot · Answer

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What is the units digit of x² - y² ?

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