Dealing With Digits

Quite a few GMAT Math questions deal in some way or other with the digits of numbers. Rather than taking a number as a whole (e.g., x = 181), it breaks the number down into its digits (e.g., "the units digit of x is 1").

The most fundamental concept concerning digits is the names of the digits themselves. The names can be somewhat confusing, so it's important to make sure you are clear on which are which.

Let's consider the number 5,182.736. Here are the names of the digits from left to right:

• 5: thousands
• 1: hundreds
• 8: tens
• 2: units (this is the name that tends to throw people off)
• 7: tenths
• 3: hundredths
• 6: thousandths

Note that while "tens" are the second number to the left of the decimal point, "tenths" are the first number to the right. Because that can get a little complicated, it's better to think about the names for what they really mean. In the number 75, there are 7 "tens." In the number 2.9, there are 9 "tenths."

Handling Variables

It's one thing to talk about the names of digits when dealing with concrete numbers, but things get more complicated when you are told, for instance, "the tens digit of x is a."

Again, it's useful to think about the relationship of digits to the numbers they form. The numbers 1, 9, and 2 are closely related to the number 192, but how?

1 is the number of hundreds, so we can start by multiplying the hundreds digit by 1. Next, 9 is the number of tens, so we multiply that by 10. 2 is the number of units, so we add that. All together: 192 = 1(100) + 9(10) + 2(1). We can extend that to numbers to the right of the decimal point, as well. For example, 19.53 is equal to 1(10) + 9(1) + 5(1/10) + 3(1/100).

Now we're set to accommodate variables. If x is a two-digit number in which the tens digit is m and the units digit is n, x = 10m + n. Using the technique described in the above paragraph, you can extend that logic to any number of digits. (Though the GMAT will rarely require that you do so with more than two or three digits.)

Digits and Number Properties

Digits questions often overlap with other number theory concepts, such as multiples and factors. For instance, the units digit determines whether an integer is a multiple of several possible factors:

• If the units digit is odd, the number is odd.
• If the units digit is even, the number is even.
• If the units digit is 0 or 5, the number is a multiple of 5.
• If the units digit is 0, the number is a multiple of 10.

It's difficult to provide a complete overview of digits on the GMAT--they can crop up in a question on just about any topic, from decimals to permutations. That said, understanding the names of the digits and how digits relate to the numbers they create will give you a head start, regardless of what else the question tests.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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