GMAT Quant Chapter 3: Properties of Numbers

Question 1

For the GMAT, is 0 considered to be positive or negative?

Answer 1

Neither.

The GMAT does not consider 0 to be positive or negative.

Question 2

What is the square root of 0?

Is 0 even or odd?

0 is the only number that …?

Answer 2

0

Even

is equal to its opposite.

Question 3

What is the number with only 1 factor?

What is the first prime number?

Answer 3

2.

Question 4

What do even exponents always produce?

Answer 4

Positive results.

Question 5

How can we determine all the factors of a number?

Answer 5

Start at 1 and list all of the factors.

Question 6

What are the first 25 prime numbers?

Answer 6

2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97

Question 7

Define a composite number.

How can we express the factors of this type of number?

Answer 7

Any number that is not prime.

As a product of its prime factors.

Question 8

How do we find the number of factors for a given number?

Answer 8

Step I. Prime Factorize
Step II. +1 to the value of each exponent of the PFs.
Step III. Multiply all of the (PFs +1) together

Question 9

What is the difference between a unique prime factor and a prime factor?

Answer 9

A unique prime factor is the only prime factor for a given number.

A prime factor is any number that is only divisible by 1 and itself.

Question 10

How do we calculate the Lowest Common Multiple (LCM) of two numbers?

Answer 10

1. Prime Factorize each integer
2. If any PFs are repeated, take only the largest exponent
3. Take all non-repeated PFs
4. Multiply the values from Steps 2 & 3

Question 11

How do we find the LCM for more than 2 integers?

Answer 11

1. PF
2. Find repeated PFs with highest exponent (PF shared by at least 2 numbers)
3. Take all non-repeated PFs
4. Multiply steps 2 & 3 values

Question 12

How do we calculate the LCM if two integers do not share PFs?

Answer 12

Multiply the integers together.

Question 13

How do we find the Greatest Common Factor (GCF)?

Answer 13

1. PF each number
2. Find repeated PFs
3. take smallest exponents of repeated PFs
4. Multiply numbers from step 3

Question 14

How do we calculate the GCF if there are no repeated PFs?

Answer 14

take only those with the smallest exponent

Question 15

What is the GCF if a set of integers has no prime factors?

Answer 15

1.

Question 16

If we have two integers x and y, and we know the LCM of one and the GCF of another, what can we calculate?

Answer 16

LCM x GCF of two seperate integers = the product of those two integers.

Question 17

What is the result when we multiply all of the unique prime factors?

Answer 17

The LCM.

Question 18

To check whether a certain number is divisible by another, what must we do?

Answer 18

Prime factorize to see if the denominator cancels out with the numerator.

Question 19

If x is divisible by y, then what must be true about the divisibility of x?

Answer 19

x must be divisible by all of the factors of y

Question 20

If z is divisible by x and y, what must be true about the divisibility of z?

Answer 20

Z must be divisible by the LCM of x and y but not higher multiples of the LCM.

Question 21

Is 0 divisible?

Answer 21

0 is divisible by any number other than itself.

Question 22

How do we know if a number is divisible by 11?

Answer 22

If the sum of the odd numbered placed digits minus the sum of the even placed digits is divisible by 11

Question 23

How do we know if a number is divisible by 12?

Answer 23

If it is both divisible by 3 and 4.

Question 24

What will the product of any n consecutive integers be divisible by?

Answer 24

n!

Question 25

what is the algebraic form of factored and non-factored consecutive integers?

Answer 25

n(n -1) n(n+1) n squared -- n n squared + n

Question 26

What is the product of n consecutive even integers divisible by?

Answer 26

2 to the power n * n!

Question 27

What is the division formula?

Answer 27

Dividend = Divisor x Quotient + remainder

Question 28

How do we calculate the remainder for two integers multiplied together?

Answer 28

calculate remainders for each integer individually, multiply the remainders together, divide result so that remainder is less than the original divisor

Question 29

Can remainders be added and subtracted?

Answer 29

Yes if we make sure the result is not larger than the divisor

Question 30

How do we calculate the number of trailing 0s for any given number?

Answer 30

PF and the number of (5x2) pairs is the number of trailing 0s.

Question 31

What do we know about a factorial greater than or equal to 5?

Answer 31

0 will always be its units digit.

Question 32

How do we calculate the total number of digits for a calculation involving massive exponents?

Answer 32

1. PF the numbers. 2. Count the number of (5x2) pairs. Each pair = one 0 3. Collect the number of unpaired 5 or 2s & any other nonzero PFs. Multiply them all together. Count number of digits in product. 4. Sum number of digits from steps 2 & 3

Question 33

How can we quickly determine how many leading 0s a decimal has?

Answer 33

If X is an integer with k digits & x is not a power of 10. Then 1/x will have (k-2) leading 0s in its decimal form.

Question 34

What is the product of any consecutive integers always divisible by?

Answer 34

Any of the integers in the set or by any of the factor combinations of the numbers

Question 35

How do you determine the number of a given prime number in a factorial?

Answer 35

Divide the integer by the PF in its progressive powers. e.g. integer / 3 = x integer / 9 = y integer / 27 = z integer / 81 = 0 The sum of x,y, and z is the number of PFs in that factorial.

Question 36

What are the first 9 perfect cubes?

Answer 36

0 1 8 27 64 125 216 343 512

Question 37

Define terminating decimal.

Answer 37

A number with a finite number of digits to the right of the decimal point.

Question 38

How do we know if a fraction produces a terminating decimal?

Answer 38

If, when factorized, the denominator contains only 2s or 5s then the decimal will terminate. A factorized denominator with any other prime factors will produce a decimal that does not terminate.

Question 39

Name the pattern of the units digits when a number is divided by the integers 1 through 9.

Answer 39

2: 2-4-8-6 Repeat 3: 3-9-7-1 Repeat 4: 4-6 Repeat 5: All end in 5 6: All end in 6 7: 7-9-3-1 Repeat 8: 8-4-2-6 Repeat 9: 9-1 Repeat

Question 40

What do we know about 2 consecutive integers and their PFs?

Answer 40

They will never share the same prime factors.

Question 41

What is the GCF of two consecutive integers?

Answer 41

1