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

How do you tell if an integer is divisible by 2?

by 3?

by 4?

by 5?

by 6?

by 8?

by 10?

Answer 16

2: It’s even
3: The sum of the digits is a multiple of 3
4: Divisible by 2 twice, or the last 2 digits are divisible by 4
5: ends in 0 or 5
6: Divisible by both 2 and 3
8: Divisible by two 3 times, or the last 3 digits are divisible by 8
10: ends in 0

Question 17

Answer 17

Question 18

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

Answer 18

X must be a multiple of 4

“Even” means a multiple of 2

x/2 = even

x = 2 * even

x = multiple of 4

Question 19

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 19

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 20

What is the Least Common Multiple of 18 and 24?

Answer 20

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 = 2³ * 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 21

Answer 21

Question 22

Data Sufficiency:

Answer 22

Question 23

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

Answer 23

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 24

What is the prime factorization of 360?

(Draw a factor tree and then write the exponents)

Example:

Answer 24

Question 25

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

Answer 25

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

The image below shows an example of 15,000:

Question 26

What is the remainder for 75/8?

What is 75/8 as a mixed number?

Answer 26

Remainder is 3. 9 ^{3 /}₈

The remainder is the number left over after it is divided evenly.

If there were 75 cookies divided evenly among 8 people, each person would get 9, with 3 left over.

The 3 left over is 3/8 for each person.

Question 27

When positive integer x is divided by positive integer y, the remainder is 2.

If x/y = 6.4, what is y?

Answer 27

5

Translate 1st sentence: x/y = integer + 2/y

x/y = 6 + .4

The decimal .4 is equivalent to the Remainder / y

.4 = 2/y

y = 2/.4 = 5

e.g. x = 32 cookies divided by y = 5 people

x/y = 32/5 = 6.4 = 6 each with 2 left over (remainder of 2)

2/y represents the 2/5 of a cookie for each person left over

Question 28

If x is the product of the integers from 1 to 10, inclusive, what is the greatest integer n for which 2ⁿ is a factor of x?

Answer 28

8

We can represent x as 10! (10 Factorial)

x = 10! = 10*9*8*7*6*5*4*3*2

The question boils down to “How many 2’s are in the prime factorization of x?”

Only the even numbers have 2’s as factors:

2 has 1

4 has 2 (2²)

6 has 1 (2*3)

8 has 3 2’s (2³)

10 has 1 (2*5)

Add up all the 2’s: 1+2+1+3+1 = 8

Question 29

Class X has more than 15 but fewer than 30 students. If they all form teams of 5, except for one team of 2, what numbers of students are possible?

Answer 29

17, 22, 27

Set K as the # of teams of 5

We can represent students as 5K + 2

5K + 2 is 2 more than a multiple of 5: 7, 12, 17, 22, 27, 32

17, 22, and 27 are between 15 and 30

Question 30

Class X has more than 20 but fewer than 40 students. They sometimes form teams of 4, except for one team of 3. They sometimes form teams of 5, except for one team of 6. How many students are there?

Answer 30

31

Translate: Teams of 4 and one team of 3 –>

4K + 3 (with K as # of teams of 4)

3 more than multiples of 4: 23, 27, 31, 35, 39

Teams of 5 and one team of 6 –>

5J + 6 (with J as # of teams of 5)

6 more than multiples of 5: 21, 26, 31, 36 (notice that 6 more is the same as 1 more)

31 works for both

Question 31

If x is the square of an integer, what must be true about its factors?

Answer 31

The prime factors must have Even Powers

(the factors must match up in pairs)

example: 144 = 12*12 = 2⁴ * 3²

Question 32

If x is the smallest positive integer such that 45 multiplied by x is the square of an integer, then x must be?

Answer 32

5

45 * x = Integer²

“Square of an Integer” means that the Prime Factors must have Even Powers (come in pairs)

45 = 9 * 5 = 3² * 5

So, we need another 5 to pair up with the lone 5.

(45*5 = 225 = 15² = 3² * 5²)

Question 33

DS: Is x a multiple of 3?

(1) If x were divided by 2, the remainder would be 1
(2) If x - 2 were divided by 3, the remainder would be 1

Answer 33

B: (2) Alone

(1) - Insufficient. This translates to “x is 1 more than a multiple of 2”, which means “x is odd”
(2) “x - 2 is 1 more than a multiple of 3”

x - 2 = 3*Integer + 1

x = 3*Integer + 3

Yes, x must be a multiple of 3 –> Sufficient