Number Properties

Question 1

Special Integer Divisibility Rule:

-An integer is divisible by 2 if?

Answer 1

The integer is even.

Question 2

Special Integer Divisibility Rule:

-An integer is divisible by 3 if?

Answer 2

The sum of the integer’s digits is divisible by 3.

-E.g. 72 is divisible by 3 b/c the sum of its digits is 7 + 2 = 9, which is divisible by 3.

Question 3

Special Integer Divisibility Rule:

-An integer is divisible by 4 if?

Answer 3

The integer is divisible by 2 twice, or if the last two digits are divisible by 4.

• E.g. 28 is divisible by 4 b/c you can divide it by 2 twice and get an integer result (28 ÷︎ 2 = 14 and 14 ÷︎ 2 = 7).
• E.g. For larger numbers, check only the last 2 digits; 23,456 is divisible by 4 b/c 56 is divisible by 4.

Question 4

Special Integer Divisibility Rule:

-An integer is divisible by 5 if?

Answer 4

The integer ends in 0 or 5.

-E.g. 75 and 80 are divisible by 5, but 77 and 83 are not.

Question 5

Special Integer Divisibility Rule:

-An integer is divisible by 6 if?

Answer 5

The integer is divisible by both 2 and 3.

-E.g. 48 is divisible by 6 since it is divisible by 2 (it ends w/ an 8, which is even) AND by 3 (4 + 8 = 12, which is divisible by 3).

Question 6

Special Integer Divisibility Rule:

-An integer is divisible by 8 if?

Answer 6

The integer is divisible by 2 three times, or if the last three digits are divisible by 8

• E.g. 32 is divisible by 8 since you can divide it by 2 three times and get an integer result (32 ÷ 2 = 16, 16 ÷ 2 = 8, and 8 ÷ 2 = 4).
• E.g. for larger numbers, check only the last three digits; 23,456 is divisible by 8 b/c 456 is divisible by 8.

Question 7

Special Integer Divisibility Rule:

-An integer is divisible by 9 if?

Answer 7

The sum of the integer’s digits is divisible by 9.

-E.g. 4,185 is divisible by 9 since the sum of its digits is 4 + 1 + 8 + 5 = 18, which is divisible by 9.

Question 8

Special Integer Divisibility Rule:

-An integer is divisible by 10 if?

Answer 8

The integer ends in 0.

-E.g. 670 is divisible by 10.

Question 9

Special Integer Divisibility Rule:

-Which integers have special divisibility rules?

Answer 9

• Divisibility rules are shortcuts to determine whether an integer is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
• Note: no rule exists for divisibility by 7 - simplest way to check for divisibility by 7 or by any other number not found in this list is to perform long division.

Question 10

Factor Pairs:

-What is a factor pair?

Answer 10

Factor pairs for any integer are the pairs of factors that, when multiplied together, yield that integer.

-E.g. Factor pairs of 8 are (1,8) and (2,4).

Question 11

Factor Pairs:

-Find the factor pairs of 72

Answer 11

Steps:

1. Make a table w/ 2 columns labelled Small and Large.
2. Start w/ 1 in the Small column and 72 in the Large column.
3. Test the next possible factor of 72 (which is 2). Divide 72 by 2 to find the factor pair: 36.
4. Repeat this process until the numbers in the Small and Large columns run into each other. In this e.g., when you have tested 8 and found that 9 is its paired factor, you can stop.

Small Large

1 72

2 36

3 24

4 18

6 12

8 9

Question 12

What is the rule regarding adding or subtracting multiples of N, where N is an integer?

Answer 12

If you add or subtract multiples of N, the result is a multiple of N.

Same as saying if N is a divisor of x and y, then N is a divisor of x + y.

E.g. By adding two multiples of 7, you get another multiple of 7:

35 + 21 = 56; (5 x 7) + (3 x 7) = (5 + 3) x 7 = 8 x 7

E.g. By subtracting two multiples of 7, you get another multiple of 7:

(5 x 7) - (3 x 7) = (5 - 3) x 7 = 2 x 7

Question 13

1. Define a prime number.
2. List the first ten prime numbers.

Answer 13

1. A prime number is any positive integer with exactly two different factors: 1 and itself. In other words, a prime number has no factors other than 1 and itself.

E.g. 7 is a prime number because the only factors of 7 are 1 and 7. However, 8 is not prime b/c it is divisible by 2 and 4.

Note: 1 is not prime, as it has only one factor (itself). Also, 2 is the only even prime.

1. The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

Note: Memorize the first ten prime numbers - will save you time on the GMAT.

Question 14

How can breaking a number down to its prime factors (i.e. prime factorization) help you determine all the factors of that number?

Answer 14

Once you know the prime factors of a number, you can determine all the factors of that number, even large numbers. The factors can be found by building all the possible products of the prime factors.

Question 15

What is the factor foundation rule?

Answer 15

The GMAT expects you to know the factor foundation rule: if a is a factor of b, and b is a factor of c, then a is a factor of c. In other words, any integer is divisible by all of its factors - and it is also divisible by all of the factors of its factors.

E.g. Since 12 is a factor of 72, then all of the factors of 12 are also factors of 72.

Question 16

What is a prime box? What is the prime box used for?

Answer 16

A prime box is a box that holds all the prime factors of a number (i.e. the lowest-level building blocks).

• Notice that you must repeat copies of the prime factors if the number has multiple copies of that prime factor.
• You can use the prime box to test whether or not a specific number is a factor of another number.

E.g. of prime boxes for 72, 12 and 125:

Question 17

Given that the integer n is divisible by 8 and 15, is n divisible by 12?

Answer 17

1. Factor both numbers: 8 = 2 x 2 x 2 and 15 = 3 x 5
2. Although you don’t know what n is, n has to be divisible by any number made up of those primes.
3. Build the partial prime box of n.
4. Because 12 = 2 x 2 x 3, then yes, n is also divisible by 12.

Notice the ellipses and question mark (“… ?”) in the prime box of n. This indicates that you have created a partial prime box. Instead of a complete set of prime factors of n, you only have a partial list of prime factors of n, because n is an unknown number. You know that n is divisible by 8 and 15, but you do not know what additional primes, if any, n has in its prime box.

Most of the time:

• When building a prime box for a variable, you will use a partial prime box.
• When building a prime box for a number, you will use a complete prime box.

Question 18

Define the following terms:

• dividend
• divisor
• quotient
• remainder

Answer 18

Every division has four parts:

• The dividend is the number being divided.
  
  • In 8 ÷ 5, the dividend is 8.
• The divisor is the number that is dividing.
  
  • In 8 ÷ 5, the divisor is 5.
• The quotient is the number of times that the divisor goes into the dividend completely. The quotient is always an integer.
  
  • In 8 ÷ 5, the quotient is 1 because 5 goes into 8 one time completely.
• The remainder is what is left over if the dividend is not divisible by the divisor.
  
  • In 8 ÷ 5, the remainder is 3 b/c 3 is left over after 5 goes into 8 once.

Putting it all together, you have 8 ÷ 5 = 1, with a remainder of 3.

This relationship can be expressed as a general formula:

Dividend = Quotient x Divisor + Remainder

(or, Dividend = Multiple of Divisor + Remainder)

Question 19

Arithmetic Rules of Odds and Evens

What are the rules of addition and subtraction with odd numbers and even numbers, i.e.:

Even ± Even = ?

Odd ± Odd = ?

Even ± Odd = ?

Answer 19

Addition and Subtraction:

If they’re the same, the sum or difference will be even. If they’re different, the sum or difference will be odd.

Even ± Even = Even

• E.g. 8 + 6 = 14

Odd ± Odd = Even

• E.g. 7 + 9 = 16

Even ± Odd = Odd

• E.g. 7 + 8 = 15

Question 20

Arithmetic Rules of Odds and Evens

What are the rules of multiplication and division with odd numbers and even numbers, i.e.:

Multiplication:

Even x Even = ?

Odd x Odd = ?

Even x Odd = ?

Division:

Even ÷​ Even = ?

Odd ÷ Odd = ?

Even ÷ Odd = ?

Answer 20

Multiplication:

Rule 1: If one even number is present, the product will be even. If you have only odd numbers, the product will be odd.

Even x Even = Even

• E.g. 2 x 4 = 8

Odd x Odd = Odd

• E.g. 3 x 5 = 15

Even x Odd = Even

• E.g. 4 x 3 = 12

Rule 2: If you multiply together several even integers, the result will be divisible by higher and higher powers of 2, b/c each even number will contribute at least one 2 to the factors of the product.

• E.g. 2 even integers in a set of integers being multiplied together:
  
  • 2 x 5 x 6 = 60 (divisible by 4)
• E.g. 3 even integers in a set of integers being multiplied together:
  
  • 2 x 5 x 6 x 10 = 600 (divisible by 8)

Division:

There are no guaranteed outcomes in division, b/c the division of two integers may not yield an integer result. In these cases, you’ll have to try the actual numbers given. You can use the special integer divisibility rules to help determine the outcome.

Question 21

When a GMAT question gives you a variable x, should you assume that x is a positive number?

Answer 21

No. Note that a variable (such as x) can have either a positive or a negative value, unless there is evidence otherwise. The variable x is not necessarily positive, nor is -x necessarily negative.

• E.g. if x = -3, then -x = 3.

Question 22

What is an absolute value of a number?

Answer 22

The absolute value of a number answers this question: how far away is the number from 0 on the number line?