What is the units digit of x² - y² ?

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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

Answer: C
Source: gmatprepnow.com
Brent Hanneson - Creator of GMATPrepNow.com
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GMAT/MBA Expert

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[email protected] wrote:
Mon Nov 21, 2022 7:16 am
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

Answer: C
Source: gmatprepnow.com
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) = 27, 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) = 33, 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) = 20, 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) = 95, 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
Brent Hanneson - Creator of GMATPrepNow.com
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