Number Properties

The SBC GMAT Files

Number Properties:
Staff, Test Prep New York

Let’s start with the tough news today. The tricky part about GMAT questions on number properties is the fact that they can take multiple forms. They are often disguised, use different terminologies, have traps, or are based on common math errors. Number properties are the building blocks for a variety of questions.

Now for the good news.

Even though these questions are often labeled ‘difficult’, they are essentially based on a set of ‘properties’. No matter the twisting, turning, disguising, or whatever the test taker has fancied, she has to base the question on ‘something’! That something is nothing more dreadful than a set of basic properties. Remember in school how they asked you a True/False question and included a ‘no’ in it? Something like, “Zero is not a negative number.” This statement is entirely true since Zero is neither a positive nor negative number. The ‘not’ and ‘negative’ in the sentence are used simply to confuse the student ”“ or more aptly – to separate the average from the smart ones.

Number Properties questions are sophisticated versions of such roundabouts. Really, that’s all.

How do you tackle this?

Step 1: Master the definitions. Know them like you know your name. Here is a list. Try to articulate each of the following list of definitions, what each means, how one is different from another, and which one includes another. For instance, all whole numbers are integers but not all integers are whole numbers.

Whole Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers, Imaginary Numbers, Factors, Divisors (Gah, these two mean the same. I was just checking!), Multiples, Parity (Wow, is that a new word to you?), Greatest Common Divisor, Least Common Multiple, Set, Absolute Value of a Number.
Set: A set is a collection of well-defined objects. Thus, a number set is a collection of well-defined numbers. In turn, a ‘set of consecutive integers’ is self-explanatory and well defined whereas ‘a set of numbers’ is not well-defined and can be anything like {1, 2, 3} or {3, 17, 98, 0.97}. The latter is not well defined.

Step 2: Master some basic properties: Know them like you know your best friend’s name. Here’s a good list to memorize:

1. Divisibility Rules:

Terms Definitions
*DIVISIBLE BY 2 IF *INTEGER IS EVEN
*DIVISIBLE BY 3 IF *SUM OF INTEGERS DIVISIBLE BY 3
*DIVISIBLE BY 4 IF DIVISIBLE BY 2 TWICE OR IF LAST TWO DIGITS DIVISIBLE BY 4
*DIVISIBLE BY 5 IF *ENDS IN 0 OR 5
*DIVISIBLE BY 6 IF INTEGER DIVISIBLE BY BOTH 2 AND 3
*DIVISIBLE BY 7 *IF CAN BE DIVIDED INTO SEVEN ”“ there is no ”˜rule’ here
*DIVISIBLE BY 8 IF *DIVISIBLE BY 2 THREE TIMES OR LAST THREE DIGITS DIVISIBLE BY 8
*DIVISIBLE BY 9 IF *SUM OF INTEGERS DIGITS DIVISIBLE BY 9
*DIVISIBLE BY 10 IF *ENDS WITH 0

2. Operating with Positive and Negative Signs:

Multiplication:(-) * (+) = – or (+) * (-) = – (order doesn’t matter)

(-) * (-) = +

(+) * (+) = +

Division:(-) ÷ (+) = – or (+) ÷ (-) = –

(-) ÷ (-) = +

(+) ÷ (+) = +3. Adding Even and Odd Numbers:

e + o = o

e + e = e

o + o = e

When in doubt: put in easy numbers to figure it out!

Step 3: Practice: This is probably an over-used but less-understood word and that is why we will elaborate on ‘how do I practice?’

When you see a question and know that it is based on Number Properties, ask a few wise questions like these:

  • Which number property is this question testing?
  • What is the property/rule? Is the question combining two properties?
  • What are the exceptions to that rule?
  • What are the common errors on the GMAT ”“ and what are the GMAT’s favorite traps?
  • Do you need to solve? If yes, up to what level – until you reach the exact answer, or half-way and then look at the answer choices? Or are you able to guesstimate?
  • How much time can you devote to this demon?! — This one is important. Remember that the GMAT is a time-based test and the test makers leverage this, which may waver your calm during a deadline-focused task.

Step 4: Maintain a Trap List and Trick List: Everyone has different weak areas in math and makes different mistakes repeatedly. Maintain a personal Trap List. Just a simple list of the traps and what you need to do to avoid being trapped by them: answer this:
“When I see X, I will think of and avoid doing Y.”

After a few weeks or months of practice, you will have transformed this weakness into a honed skill.

A Trick List is a list of your own favorite tricks: Here are ours:

The units digit of the product of two large numbers (13,877 and 89) will be the units digit of 7*9, which is 3 since 7*9 = 63. You’d use this trick when question seems to require lengthy multiplication. All you have to do is multiply the units digit and look at the answer options. It is likely you’ll be able to eliminate the majority of answers. Likewise, you can multiply the first two numbers: 1 and 8, to know that the answer will be 8 or higher – – it could even be 10, 11 or 12. Just actively engage with the question, and you can guestimate – – especially when you’re running out of time on the test!

Time to get rocking and rolling with an example!

A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

a) 13

b) 59

c) 35

d) 37

e) 12

Let’s break this beast into pieces

First, ask yourself the questions we listed prior:

1) What number property is this testing? – Divisors
2) Why does it look complicated? – Because it is asking for divisors of a remainder, not the original number.
3) What strategy looks best? – (a) Play with variables (b) When it seems easier, plug in answer choices (plugging in or substitution if answer choices contain variables and SWA (Solving with Answers) if the choices are numbers).

Let’s start with strategy (a)
Let the original number be ”˜a’; let the divisor be ”˜d’’ let the quotient of the division of ”˜a’ by ”˜d’ be ”˜x’.
So, a/d = x and the remainder is 24. Or, a= dx + 24
When twice the original number is divided by d, 2a is divided by d. We know that a = dx + 24. Therefore, 2a = 2dx + 48.

Basically, (2dx +48)/ d leaves a remainder of 11.

2dx is perfectly divisible by d and therefore will leave no remainder. The remainder 11 was obtained by dividing 48 by d. When 48 is divided by 37, the remainder is 11. Hence the divisor is 37. Viola, option D.

With Number Properties questions, another useful technique is ‘plugging numbers’. Pick a few
numbers to plug into the equation or inequality and then solve it. Picking numbers wisely like one
positive number, one negative number, a really large number and once with zero will give you an
idea about how the equation or inequality will behave.

Sometimes plugging numbers is a backup plan for DS questions because it may be a tedious process
and you may wonder if you missed an ”˜exception’ to the rule. Other times, however, it can lead you to an answer very quickly. Let’s see an example with the plugging numbers technique:

Example:

If m is an odd integer, which one of the following is an even integer?

a) 5m + 4

b) m/2

c) 2m + 1

d) m – 2

e) m(m+1)

Solution:
”˜m’ is an odd integer, so we choose an odd integer for m say 3 and ”˜plug it’ into each of the options.

a) 5m + 4 = 5 (3) + 4 = 19 = which is an odd integer, therefore A is incorrect

b) m/2 = 3/2 = 1.5 which is not an integer, let alone being an even or odd integer

c) 2m + 1 = 2 (3) + 1 = 7 which is not an even integer either

d) m -2 = 3 – 2 = 1, which is an odd integer, so eliminate this choice

e) m(m + 1) = 3 (3+1) = 12 which is an even integer. So, this is the correct option.

The most important thing to remember with plugging in, is to make sure the number you choose for plugging in has the properties given in the question.

Since number properties are the foundations on which much of the math you’ll see on the test is based, it’s important to gain a competence, even fluency with them. It should be the first thing you study”¦so get started!

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