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

Question

A and B are 2-digit numbers with non-zero digits. Both digits in A are distinct, and B is formed by reversing the digits of A. Which of the following is always a factor of  A^2 – B^2 ?

A.  5
 B.  15
 C.  45
 D.  75
 E.  99

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BrentGMATPrepNow · Accepted Answer

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)    
= (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

Chethan92 · Answer

Let the two digits in A be x and y. Then A = 10x+y.
B = 10y+x.

A^2-B^2 = 100x^2+y^2+20xy-100y^2-x^2-20xy = 99x^2-99y^2 = 99(x^2-y^2)

So  A^2-B^2  will always be a factor of 99.

E is the answer.

gracie · Answer

21^2-12^2=297
99
E

EgmatQuantExpert · Answer

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) =    
 • Or,  A^2 – B^2 =     = 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

nyadavalwar · Answer

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

BrentGMATPrepNow · Answer

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

3mrecan · Answer

A => 10A+B
B => 10B+A
(A-B).(A+B) = (9(A+B)).(11(A+B))=(A+B)(9x11)=99(A+B) must be divisible by 99.

egmat · Answer

Hi! Great question about the testing strategy. Let me clarify why this approach is actually   systematic, not luck .

Key Insight:  When using the testing method to eliminate wrong answers, you don't need to find ALL factors - you just need  ONE  example that doesn't have a particular factor to eliminate that choice.

Why 21 works well: 
• A =  21  gives us A2 - B2 =  212  -  122  =  297 
•  297  =  3  ×  99  =  3  ×  9  ×  11 
• This immediately eliminates options containing  5  (choices A, B, C, D)

The smart testing strategy: 
1. Pick numbers that avoid  5  and  0  in the digits (like  21 ,  31 ,  42 )
2. This ensures the result won't be divisible by  5 , helping eliminate multiple choices at once
3. Since choices A through D all contain  5  as a factor, one good test case eliminates them all!

General principle:  For "which is ALWAYS a factor" questions, test with values that  avoid  the factors in wrong answer choices. This makes elimination faster.

Note that  the algebraic approach (A2 - B2 =  99 (a+b)(a-b)) is more reliable, but smart testing can be equally fast in certain questions when you understand the strategy!

Answer: E

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