BTGmoderatorDC wrote:If two 2-digit positive integers have their respective tens digits exchanged, the difference between the pair of integers changes by 4. What is the greatest possible difference between the original pair of integers?

A) 76
B) 80
C) 82
D) 90
E) 94

OA C

Source: GMAT Prep

Say the first 2-digit integer is [xy] = 10x + y and the second 2-digit integer is [ab] = 10a + b such that x > a; thus, [xy] > [ab].

Difference between the original pairs = [xy] - [ab] = 10x + y - 10a - b = 10(x - a) + y - b;

After exchanging the tens digits of the integers, we have new set of integers as 10a + y and 10x + b, where 10x + b > 10a + y

Difference between the changed pairs = [xb] - [ay] = 10x + b - 10a - y = 10(x - a) - y + b;

Difference between the pair of integers = [10(x - a) + y - b] - [10(x - a) - y + b] = 2(y - b) = 4 (given)

=> y - b = 2

Note that y and b are units digits and they can have any values such that the difference between them is 2. Example: y = 2 and 0; y = 3 and 1; ... y = 9 and 7. Also, note that tens digits of the integers can be any numbers (except 0). Since we want the greatest possible difference between the original pair of integers, let's take x = 9 and a = 1.

"¢ Taking x = 9, a = 1, y = 9 and b = 7, we get [xy] = 99 and [ab] = 17. Note that you may choose any values of y and b as they can have any values such that the difference between them is 2, as discussed above.

Difference of 99 and 17 = 99 - 17 = 82 (Correct answer: Option C)

Let's cross-check this.

The new pairs of two 2-digit integers after exchanging the tens digits are 97 and 19. The difference between them = 97 - 19 = 78

We see that the difference between the pair of integers 82 - 78 changes by 4 (given).

The correct answer: C

Hope this helps!

-Jay
_________________
Manhattan Review GRE Prep

Locations: GRE Classes Seattle | GMAT Prep Course Hong Kong | GRE Prep San Francisco | SAT Prep Classes NYC | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.

BTGmoderatorDC wrote:If two 2-digit positive integers have their respective tens digits exchanged, the difference between the pair of integers changes by 4. What is the greatest possible difference between the original pair of integers?

A) 76
B) 80
C) 82
D) 90
E) 94

OA C

Source: GMAT Prep

We can let the first integer be 10a + b and the second be 10c + d originally (where a > c and hence 10a + b > 10c + d). Thus the new pair become 10c + b and 10a + d, respectively (notice that now we have 10a + d > 10c + b. From the information given in the problem, we have:

(10a + b) - (10c + d) = (10a + d) - (10c + b) + 4

10a + b - 10c - d = 10a + d - 10c - b + 4

b - d = d - b + 4

2b - 2d = 4

b - d = 2

We see that the difference between their units digits is 2 regardless of their tens digit. Thus we can let their tens digits be as far apart as possible, i.e. one of them is 1 and the other is 9.

For example, if we let the two numbers be 92 and 10, we see that the original difference is 92 - 10 = 82 and the new difference is 90 - 12 = 78 (notice that 82 is 4 more than 78).

We also could use 99 and 17. We see that the original difference is 99 - 17 = 82 and the new difference is 97 - 19 = 78. Therefore, the greatest possible difference between the original pair is 82.

Answer: C

Hi All,

We're told that if two 2-digit positive integers have their respective TENS digits exchanged, the difference between the pair of integers changes by 4. We're asked for the GREATEST possible difference between the original pair of integers. There are a few different ways to approach this question (and some of them are incredibly "step heavy"). Thankfully, we can solve it rather easily by TESTing THE ANSWERS.

Since we're looking for the GREATEST possible difference between the two 2-digit integers, we should start with Answer E. However, we can use some Number Properties to quickly eliminate a couple of the possibilities first.

The smallest 2-digit number is 10 and the largest is 99, so the difference between those two numbers is 89 (and can only get smaller). Thus, Answers D and E are NOT possible, so we do not have to consider them. Let's TEST Answer C first....

Answer C: 82
Let's pick two 2-digit numbers that have a difference of 82... the easiest would be 10 and 92.
We already know that the difference between these numbers is 82.
When we switch the TENS digits, we have 90 and 12. The difference between these numbers is 90 - 12 = 78.
The two results (82 and 78) differ by 4 - and this is an exact match for what we were told, so this MUST be the answer.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich