A and B are 2-digit numbers with non-zero digits. Both digits in A are

Solution

Given:
In this question, we are given

• A and B are 2-digit numbers, with non-zero digits.
• Both digits in A are distinct.
• B is formed by reversing the digits of A.

To find:

• Among the given options, which number is always a factor of \(A^2 – B^2\).

Approach and Working:
Let us assume that A = pq and B = qp.

• Therefore, the value of A = 10p + q
• And, value of B = 10q + p

Now, \(A^2 – B^2 = (A + B) (A – B) = [(10p + q) + (10q + p)][ (10p + q) – (10q + p)]\)

• Or, \(A^2 – B^2 = [11 (p + q)] [9 (p – q)] = 99 (p + q) (p – q)\)

Now, irrespective of the values of p and q, the expression 99 (p + q) (p – q) is always divisible by 99.

Hence, the correct answer is option E.

Answer: E

A fast approach is to test some values of A and B

For example, it COULD be the case that A = 21 and B =12
In this case, A² - B² = 21² - 12²
= (21 + 12)(21 - 12) [aside: since 21² - 12² is a DIFFERENCE OF SQUARES, we can first factor it before evaluating it]
= (33)(9)
= (3)(11)(3)(3)

Since 5 is not a factor of (3)(11)(3)(3), we can eliminate answer choice A
Since 15 is not a factor of (3)(11)(3)(3), we can eliminate answer choice B
Since 45 is not a factor of (3)(11)(3)(3), we can eliminate answer choice C
Since 75 is not a factor of (3)(11)(3)(3), we can eliminate answer choice D
HOWEVER, 99 IS a factor of (3)(11)(3)(3). So, KEEP answer choice E

Answer: E

Cheers,
Brent

Although this is definitely a fast approach, it seems like pure luck. The algebraic approach only took 30-40 seconds, and I think if we go down guessing from such vast combinations, we will waste more time.
Did you make a calculated guess of putting 21? If yes, then please explain, it'd be helpful

No, I didn't make a calculated guess by starting with 21.
I chose that number because 21 and 12 result in pretty small numbers when squared.

Let's try 31 and 13
31² - 13² = 792

792 = (2)(2)(2)(3)(3)(11)
Looks like only 99 works here as well.

What about 32² - 23² = 495
495 = (5)(9)(11)
This time A, B, C and E work (which means I'd have to go back and test another pair of values)

So, sometimes testing answer choices will yield the correct answer immediately, and other times it won't.

Keep in mind that, when it comes to answering GMAT questions, the best approach is the fastest approach.
If you have the mindset that testing answer choices is an inferior approach, you'll miss out on an important strategy.

Cheers,
Brent

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