[DS]If a, b, and c are integers and abc ≠0

[acchi369]

If a, b, and c are integers and abc ≠ 0, is a2 - b2 a multiple of 4?

(1) a = (c - 1)^2

(2) b = c2 - 1

Answer= C

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I have a question on this problem. If you were to multiply out statement 1, you would get:

a = (c-1)^2
a = (c-1)(c-1)
a = c^2 -2c + 1

So, if you were to plug that in for a in the original equation, you would get:

a^2 - b^2
(a+b)(a-b)
((c^2 -2c + 1) + b)((c^2 -2c + 1) -b)

So, without solving that equation, wouldn't you know that the term has to be divisible by 4 given that the middle term -2c is in there twice so 2c * 2c is 4c^2?

I am probably making the wrong assumption but I feel like that was enough information for me.

[srcc25anu]

See image attached for detailed calculations.

Ans C

[acchi369]

See image attached for detailed calculations.

Ans C

Thanks for the reply. I guess I understand the principle but is their a way of testing those different outcomes without picking numbers and trying each possibility to see if it fits the rules?

[Brent@GMATPrepNow]

a^2 - b^2
(a+b)(a-b)
((c^2 -2c + 1) + b)((c^2 -2c + 1) -b)

So, without solving that equation, wouldn't you know that the term has to be divisible by 4 given that the middle term -2c is in there twice so 2c * 2c is 4c^2?

Sure, the middle term might be divisible by 4, but we cannot conclude that the entire expression is divisible by 4. For example, 20 is divisible by 4, but 1 + 20 + 3 is not divisible by 4.

Cheers,
Brent

[Brent@GMATPrepNow]

If a, b, and c are integers and abc ≠ 0, is a^2 - b^2 a multiple of 4?

(1) a = (c - 1)^2

(2) b = c^2 - 1

NOTE: I added some exponent signs (^) to the question to avoid any confusion.

Target question: Is a^2 - b^2 a multiple of 4?

Since a^2 - b^2 = (a + b)(a - b), we can rephrase the target question . . .

Rephrased target question: Is (a + b)(a - b) a multiple of 4?

Statement 1: a = (c - 1)^2
This tells us a little bit about the properties of a, but we know nothing about b. So, there is no way to determine whether or not (a + b)(a - b) is a multiple of 4.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b = c^2 - 1
This tells us a little bit about the properties of b, but we know nothing about a. So, there is no way to determine whether or not (a + b)(a - b) is a multiple of 4.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
Statement 1 tells us that a = (c - 1)^2
Expand to get: a = c^2 - 2c + 1

Statement 2 tells us that b = c^2 - 1

No we can replace a and b with their respective values:
(a + b)(a - b) = [(c^2 - 2c + 1) + (c^2 - 1)] [(c^2 - 2c + 1) - (c^2 - 1)]
= [2c^2 - 2c][-2c + 2]
= [2(c^2 - c)][2(-c + 1)] I factored the 2 out of each set of brackets
= 4(c^2 - c)(-c + 1)
At this point, we can see that (a + b)(a - b) is definitely a multiple of 4.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C


Cheers,
Brent

[faraz_jeddah]

If a, b, and c are integers and abc ≠ 0, is a^2 - b^2 a multiple of 4?

(1) a = (c - 1)^2

(2) b = c^2 - 1

NOTE: I added some exponent signs (^) to the question to avoid any confusion.

Target question: Is a^2 - b^2 a multiple of 4?

Since a^2 - b^2 = (a + b)(a - b), we can rephrase the target question . . .

Rephrased target question: Is (a + b)(a - b) a multiple of 4?

Statement 1: a = (c - 1)^2
This tells us a little bit about the properties of a, but we know nothing about b. So, there is no way to determine whether or not (a + b)(a - b) is a multiple of 4.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: b = c^2 - 1
This tells us a little bit about the properties of b, but we know nothing about a. So, there is no way to determine whether or not (a + b)(a - b) is a multiple of 4.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Statements 1 and 2 combined:
Statement 1 tells us that a = (c - 1)^2
Expand to get: a = c^2 - 2c + 1

Statement 2 tells us that b = c^2 - 1

No we can replace a and b with their respective values:
(a + b)(a - b) = [(c^2 - 2c + 1) + (c^2 - 1)] [(c^2 - 2c + 1) - (c^2 - 1)]
= [2c^2 - 2c][-2c + 2]
= [2(c^2 - c)][2(-c + 1)] I factored the 2 out of each set of brackets
= 4(c^2 - c)(-c + 1)
At this point, we can see that (a + b)(a - b) is definitely a multiple of 4.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C


Cheers,
Brent

Hi Brent. Typo in your solution. S1 and S2 independently are INsufficient.

[Brent@GMATPrepNow]

Hi Brent. Typo in your solution. S1 and S2 independently are INsufficient.

Thanks for catching that, faraz.
I've edited my response.

Cheers,
Brent