Welcome back to our series on number properties. In the last article, we learned about place-value problems. This article will cover the related topic of switching digits in two-digit integers. Here are some examples of what that means:

Place-value problems

75 <-> 57

39 <-> 93

62 <-> 26

Some GMAT quant problems ask about the difference (in terms of subtraction) between two such numbers, or about the difference between the individual digits used to build these numbers.

Here’s an official GMAT problem:

If a two-digit 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

These problems can be baffling the first time you see one, but there is a straightforward rule that makes them very easy.

The difference between the two-digit integers xy and yx is 9 times the difference between the digits x and y.

Given this rule, all you need to do on the problem above is take 27/9 = 3, and the correct answer choice is A.

Here are some examples of digit switching, to be followed by an explanation of the rule.

  • 75 <-> 57

          75 – 57 = 18

          7 – 5 = 2

          18 / 2 = 9

  • 39 <-> 93

          93 – 39 = 54

          9 – 3 = 6

          54 / 6 = 9

  • 62 <-> 26

          62 – 26 = 36

          6 – 2 = 4

          36 / 4 = 9

Why does it work this way? The answer has everything to do with the way our number system is built. In any number, the tens digit “counts” ten times as much as the units digit. When switching digits of a two-digit number, each digit changes by the same amount but in opposite directions. If the tens digit increases by 2, the units digit decreases by 2. If the tens digit decreases by 5, the units digit increases by 5. But the change for the tens digit “counts” ten times as much as the change for the units digit.

75 <-> 57

Here the digits differ by 2. If the 7 in the tens place becomes a 5, the value decreases by 20. Meanwhile, the 5 in the units place becomes a 7, and the value increases by 2. So when the digits switch places, the change in the units place always “undoes” 1/10 of the change in the tens place, resulting in a “net” change that is always a multiple of 9.

You can practice switching digits with any two-digit number to watch this property at work. And whenever you see it on GMAT quant, you’ll be ready.

In the next article, we’ll consider the “spacing” of multiples of a given integer and how this spacing relates to series of consecutive integers.

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Contributor: Elijah Mize (Apex GMAT Instructor)