Factors & Multiples

Question 1

Factors

Answer 1

Positive integers that can divide N into another integer

Question 2

Multiples

Answer 2

Integers that result from multiplying N by other integers (including 0)

Question 3

Prime

Answer 3

A positive integer with exactly two distinct factors (itself and 1)

Question 4

Divisibility

Answer 4

x is divisible by y if y is a factor of x

Question 5

Zero

Answer 5

0 is an even integer (but is neither positive nor negative). 0 is a multiple of everything (don’t forget this when counting possible multiples!)

Question 6

One

Answer 6

1 is not a prime number (it only has one factor: “itself”). 1 is a factor of everything (don’t forget this when counting factors!)

Question 7

Two

Answer 7

2 is the only even prime number and the lowest prime number

Question 8

Divisibility rules for 2, 3, 4, 5, 6, 9, 10

Answer 8

• 2 if the number is even
• 3 if the sum of the digits is a multiple of 3 (e.g. 327: Sum of 3 + 2 + 7 = 12. 327/3 = 109.)
• 4 if the last two digits form a multiple of 4 (e.g. 2364: 64 is a multiple of 4, so 2364 is divisible by 4.)
• 5 if the last digit is 0 or 5
• 6 if the rules for both 2 and 3 are met
• 9 if the sum of the digits is a multiple of 9 (e.g. 288: Sum of 2 + 8 + 8 = 18. 288/9 = 32.)
• 10 if the last digit is 0

Question 9

Least Common Multiple

Answer 9

Least Common Multiple: The LCM is the smallest positive multiple shared by two or more integers. To find the LCM, reduce each element to its prime factorization; the LCM is the product of the highest counts of each separate prime.

Example: Find the Least Common Multiple of 8 and 18.

LCM requires three 2s and two 3s:

Or find it on the number line:
Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80…
Multiples of 18 = 18, 36, 54, 72, 90…

The least common multiple is 72.

Question 10

Greatest Common Factor

Answer 10

Greatest Common Factor: The largest factor shared between two (or more) integers. To find the GCF, first find the prime factorization of each integer. The GCF is the product of the lowest counts of each prime common to both numbers.

Example: Find the Greatest Common Factor of 108 and 48

108 = 3^4 * 2^2
48 = 2^4 * 3

Each term has in common two 2s and one 3, so the GCF is 2^2 * 3 = 12

My job is to find exactly what is common between the numbers.

Question 11

Notable Multiple Rule

Answer 11

Because a multiple of N is N • any integer, multiples are infinite.

Question 12

Notable Factor Rule

Answer 12

Because only so many integers divide into N, factors are finite.

Question 13

Notable Multiple Rule 2

Answer 13

Because 0 • anything is 0, 0 is a multiple of all numbers.

Question 14

Prime Rule 2

Answer 14

Prime must be positive and neither 0 nor 1 is a prime number.

Question 15

List the prime numbers 2-35

Answer 15

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

Question 16

7 Divisibility

Answer 16

Algebraic theory tells us that:
a(b+c) = ab + ac

Similarly:

7 · 136 = 7(100 + 30 + 6) = 700 + 210 + 42

Question 17

Percent problems

Answer 17

Perhaps the single greatest mistake that examinees make with percent questions is that they often take the percentage of the incorrect value. When taking a percentage, ask yourself the question “of what?” so that you organize your thoughts appropriately. Consider this example:

What is 25 increased by 20%?
25 + 20/100

The answer clearly isn’t 25.2, so what comes next?

The 20% must be a fraction of a value; in this case, it’s a fraction of itself.

25 + 20/100 · (25)