Number Properties

Question 1

Is 0 odd or even?

Is 0 positive or negative?

Answer 1

0 is even.

0 is neither positive or negative. Problems often use the phrase “positive integers”– read carefully and realize they are talking about 1, 2, 3…. etc

Question 2

What is 5 - (9 - x) ?

Answer 2

-4 + x

Be sure to distribute the - sign!

The double negative becomes a positive

5 - 9 + x

Question 3

If xy < 0 , what can we conclude about x and y?

Answer 3

x and y have opposite signs:

one is positive, one is negative

Question 4

If xyz > 0 , what can we conclude about x, y and z?

Answer 4

All 3 can be positive, OR there are 2 negatives and 1 positive

Question 5

If y is an integer, what determines whether (-2)^y is positive or negative?

Answer 5

If y is even, (-2)^y is positive

If y is odd, (-2)^y is negative

e.g. (-2)⁴ = -2 * -2 * -2 * -2 = 16

(each pair of negative signs cancels each other out)

(-2)⁵ = (-2)⁴ * -2 = 16 * -2 = -32

(there is 1 negative sign left without a pair)

Question 6

What is (-2)³ * 1/2 * -3 * -1/2

Answer 6

-6

First, determine whether it’s positive or negative

There are an ODD # of negative signs, so it will be negative.

-8 * 1/2 * -3 * -1/2 = -6

Question 7

If x^y < 0, what can we conclude about x and y?

Answer 7

x must be negative

y can be positive or negative but can’t be an even integer

(-2)³ = -8

(-2)^-3 = -1/8

(-2)⁴ = 16

Question 8

DS: If (y - x) / z < 0, is y > x?

(1) z < 0
(2) y + x < 0

Answer 8

A: (1) Alone is Sufficient

If (y - x) / z < 0, (y - x) and z must have opposite signs

(1) If z is negative, (y - x) must be positive –> YES, y > x
(2) Insufficient. We can only conclude that x and y are not both positive. We don’t know which is greater.

Question 9

Answer 9

Question 10

If x is an integer, what can we conclude about 2x?

Answer 10

2x must be even

even * any integer = even

Question 11

Answer 11

If b is odd, which must be even? E: 2b + 2

A) 2b - 3 –> 2b is even –> even - odd = odd

B) (b-1) / 2 –> even / 2 = even?

If b is a multiple of 4, yes. If not, no. (4/2 = 2 , 6/2 = 3)

C) b/2 must be a fraction, since odd means NOT a multiple of 2 (3/2 = 1.5)

D) odd*odd + even = odd + even = odd

E) even + even = even

Question 12

DS: Is xy even?

(1) y = x - 1
(2) x is an integer

Answer 12

C - Both Together are Sufficient

if either X OR Y is even, xy will be even, so it’s sufficient

1) Insufficient – we don’t know if they are integers
2) Insufficient – we don’t know anything about y

Together– we know they are Consecutive Integers, so one must be an odd integer, and the other must be an even integer

example: x = 2, y = 1 –> xy = 2

Question 13

What are the first 10 prime numbers?

Answer 13

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Question 14

DS: Is x prime?

(1) x > 2
(2) x is even

Answer 14

C: Both Together are Sufficient.

The only even prime is 2. So if x >2 and x is even, the answer to the question is “NO”. Alone each statement is insufficient.

Question 15

If x is divisible by y, how can we write that?

Answer 15

x / y = Integer

x = y * Integer

We can also say that y is a Factor of x,

and x is a Multiple of y

Question 16

Answer 16

Question 17

If x/2 is even, what can we conclude about x?

Answer 17

X must be a multiple of 4

“Even” means a multiple of 2

x/2 = even

x = 2 * even

x = multiple of 4

Question 18

Do the following have the same meaning? Or are any of them different?

a. 3 is a factor of 12
b. 3 is a divisor of 12
c. 12 is a multiple of 3
d. 12 is divisible by 3
e. 12/3 is an integer
f. 12 is equal to 3n, where n is an integer
g. 12/3 yields a remainder of 0
h. 12 items can be shared among 3 people so that each person has the same number of items

Answer 18

Yes, they all have the same meaning!

The GMAT often writes questions in different ways to confuse you.

It’s important to be able to translate the language into something you understand.

Question 19

What is the Least Common Multiple of 18 and 24?

Answer 19

We factor each number and then find the “unique factors” – we find the highest power of each different prime factor and then multiply them.

18 = 3^{<strong>2</strong>} * 2

24 = 2^{<strong>2</strong>} * 3

The Least Common Multiple is 3² * 2² = 72

We can confirm this by writing out the multiples:

18: 18, 36, 54, 72
24: 24, 48, 72

Question 20

Answer 20

Question 21

Data Sufficiency:

Answer 21

Question 22

1) If the units digit of N is 0, what do we know about the factors of N?

Answer 22

1) The factors include 2 and 5, because N must be a multiple of 10.

(For any number we want to factor that ends in one or more 0’s, it is easiest to factor the 10’s out first)

Question 23

What is the prime factorization of 360?

(Draw a factor tree and then write the exponents)

Example:

Answer 23

Question 24

If the integer N ends with “5000”, what do we know about the factors of N?

Answer 24

From the 3 zeros at the end, the factors include 3 pairs of 2’s and 5’s, because N is a multiple of 1000.

Once we factor out 1000, we know that there is another 5 as a factor, since N/1000 would end with a 5.

Factors include 2³ * 5⁴