If a two-digit positive integer has its digits reversed, the resulting

This is a tricky number properties question. Note that you can use two variables a and b to represent each of the digits.

In terms of expressing them in values - the total value would be (10a + b).

For example, for a number 37, a = 3 and b=7..then the expression 10a+b = 30 + 7 = 37.

Please refer to the video explanation here:
https://www.gmatpill.com/gmat-practice-t ... stion/2397

[youtube]https://www.youtube.com/watch?v=M9Psoio5ABQ[/youtube]

Do we use 10 because of the tens digit??

Yes, any two-digit integer ab can be expressed as 10a+b, for example: 45 = 10*4 + 5.

The same for 3, 4, 5, ... digit numbers. For example, 4-digit number 5,432 can be written as 5*1,000 + 4*100 + 3*10 + 2 = 5,432.

Hope it's clear.

The answer has to be a factor of 27, the only option that's a factor of 27 is 3.

Sice \((10x + y) - (10y + x) = 27\), you can simplify this relationship by subtracting with a common factor --> 9x - 9y = 27 ---> 9(x - y) = 27 ---> here, you already notice that the difference has to be a factor of both 9 and 27, but you can simplify further ---> x - y = 3, and thus we have the answer.

But these last steps are superfluous if you already notice that the answer has to be a factor of 27, this way you save time without having to calculate.

Hi All,

Even though this question might seem a little strange, you do NOT need to do any excessive math to get to the correct answer. With just a bit of 'playing around' you can use 'brute force' to get to the answer.

We're told that a 2-digit number has its digits reversed and the difference between those two numbers is 27.

IF we use....
11 and 11, then the difference is 0 - this is NOT a match

12 and 21, then the difference = 9 - this is NOT a match

13 and 31, then the difference = 18 - this is NOT a match (notice the pattern though? The difference keeps increasing by 9!!!!! I wonder what the next one will be???)

14 and 41, then the difference = 27 = this IS a match

The question asks for the difference in the two DIGITS. The difference between 1 and 4 is 3.

Final Answer:

There are actually several ways to get to this answer: 14 and 41, 25 and 52, 36 and 63, 47 and 74, 58 and 85, 69 and 96.

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

Let’s first label the original two-digit integer as N. We can then say that N = 10A + B, where A is the tens digit and B is the units digit of N.

If this is hard to see let’s try it with a sample number, say 24. We can say the following:

24 = (2 x 10) + 4

24 = 20 + 4

24 = 24

Getting back to the problem, we are given that if the integer N has its digits reversed the resulting integer differs from the original by 27. First let’s express the reversed number in a similar fashion to the way in which we expressed the original integer.

10B + A = reversed integer

Since we know the resulting integer differs from the original by 27 we can say:

10B + A – (10A + B) = 27

10B + A – 10A – B = 27

9B – 9A = 27

B – A = 3

Since B is the tens digit and A is the units digit, we can say that the digits differ by 3.

The answer is A.

the difference between reversed integers is always a multiple of 9
dividing that difference by 9 gives the difference between digits
27/9=3
A

If a two-digit positive integer has its digits reversed, the resulting integer differs from the original by 27. By how much do the two digits differ?

Let the 2-digit number be xy

|(10x+y) - (10y-x)| = 27
|9x-9y| =27
|x-y| = 3

IMO A

If a two-digit positive integer has its digits reversed, the resulting integer differs from the original by 27. By how much do the two digits differ?

Let the 2-digit number be xy

|(10x+y) - (10y-x)| = 27
|9x-9y| =27
|x-y| = 3


I don't understand the logic behind writing 10x instead of... 5x or 6x or whatever...Could you explain it?

Hi dodobolo,

As an example, if you take the number '85', then you can break that number down into two pieces (8)(10) and (5)(1).

Now, imagine you don't now what the two-digit number is... we can write that number as 'XY' where X represents the 'tens digit' and the Y represents the 'units digit. The value of 'XY' is (X)(10) +(Y)(1).

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

Hello Experts,
EMPOWERgmatRichC, VeritasKarishma, IanStewart, Bunuel, chetan2u, ArvindCrackVerbal, AaronPond, GMATinsight, JeffTargetTestPrep, ccooley, RonPurewal
Let,
The numbers are 96 (if the digits are being reversed then it is 69). If 69 is deducted from 96, then we get 27. But the question prompt does not specify that which one has to be deducted from which one (9-6=3 or 6-9=-3)
So, can we add one more possible choice is (E) -3?
Thanks__

Hi

No, when you talk of difference between two numbers and in the language as in this- the two digits differ by ?, the answer is always positive and we are talking of absolute differences
So 6 and 9 differ by |6-9| or 3.

Hello Asad,

If the two-digit number is AB, the reverse will be BA.
AB can be expressed as 10A + B while BA can be expressed as 10B + A.
Difference of AB and BA = (10A + B) ∆ (10B +A) = 9 (A ∆ B).
It’s given that 9 (A ∆ B) = 27. Therefore, (A ∆ B) = 3.

Asad, the word difference is always used to denote the subtraction of the smaller value from the bigger value. This is the reason why difference is always positive.
As such, the only possible value for the difference here will be 3.

Again, there was a faster way of solving this without taking values and getting confused. When you take the difference between a two-digit number and its reverse, the difference is always divisible by 9. Knowing that this difference is 27, we obtained the difference between the digits to be 3; we didn’t have to try values at all. This is not to say that the plugging in method is inferior. Only that it takes more time in problems like these.

Hope that helps!
PS: Note that I have used the symbol ∆ to represent the difference between two numbers

What if the prompt says:
If a two-digit positive integer has its digits reversed, the resulting integer differs from the original by -27. By how much do the two digits differ?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
What's the way to solve it?
Thanks__

Hello Asad,

As I discussed in my response to your previous question, difference is always positive. Therefore, there is no way the question can state the difference is -27. Difference will never be stated as negative.
At least in a good quality question!

Hope that helps!

Video solution from Quant Reasoning:
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