Factors And Multiples

The topic of this section has to do with the relationships between numbers. For example, we know that 12 and -12 are related in that they're opposites of each other.

Similarly, how do numbers like 10 and 15 relate, especially given that 10 = 5 * 2 and 15 = 5 * 3?

Divisibility

We say that integer x is divisible by integer y when:

1. y divides evenly into x, or

2. The result of the division is a whole number, or

3. We don't get a remainder from the division.

Here's some shorthand tricks for determining the divisibility of the following numbers:

• Divisible by 2 if the last digit is 0, 2, 4, 6, 8

• Divisible by 3 if the sum of the digits is divisible by 3

• Divisible by 4 if the last two digits are divisible by 4

• Divisible by 5 if the last digit is 0, 5

• Divisible by 6 if divisible by 2 and 3

• Divisible by 7 if 5 × last digit + rest of the number is divisible by 7

  • Example: Is 126 divisible by 7?

    • The last digit is 6, so 6 * 5 = 30

    • The rest of the numbers make up 12, so 12 + 30 = 42

    • 42 is divisible by 7!

• Divisible by 8 if the last three digits are divisible by 8

• Divisible by 9 if the sum of the digits is divisible by 9

• Divisible by 10 if the last digit is 0

Multiples

A multiple of x is any number that is a product of x times a whole number. For example, 2 * 3 = 6, 2 * 4 = 8, and 2 * 5 = 10, so 6, 8, and 10 are all multiples of 2.

Another way of thinking about multiples is that a multiple is divisible by its base. 10 / 2 divides evenly into the whole number 5, so 10 is a multiple of 2.

Prime and Composite

You can classify any whole number greater than 1 as either prime or composite. A prime number is only divisible by either 1 or itself. A composite number is divisible also by a whole number other than 1 or itself.

7 is a prime number. 21 is a composite number because it is divisible by 7 and 3.

Note: Another way of thinking about prime and composite is to ask if any whole number between 1 and the number in question divide evenly into the number. For example, it's a given that 1 and 5 divide evenly into 5. But what about 2, 3, and 4? Because those numbers don't divide evenly into 5, 5 is a prime number.

Prime Factorization and Product of Primes

In order to understand prime factorization and product of primes, we need to understand factors and factorization first.

Factors are basically the individual numbers in a multiplication expression. For example, 2 * 3 has factors 2 and 3. (It's like how we call the parts of an addition expression addends.)

Factorization of a whole number is the process of coming up with the whole numbers that multiply together to give us that original number. For example, we know that the factors of 12 are 1, 2, 3, 4, and 6.

Definitions

A product of primes is basically a product where every factor is a prime number. For example, a product of primes for 12 would be 2 * 2 * 3.

In the same vein, prime factorization is the process of taking a composite number and finding a product of primes for that number.

How to prime factorize

The process of breaking down a number into a product of primes, i.e., prime factorize, is to factorize your factors until they're all prime.

144 = 12 * 12
144 = (3 * 4) * (3 * 4)
144 = (3 * 2 * 2) * (3 * 2 * 2)
144 = 2^4 * 3^2

In the example above, 144 factorizes into 12 * 12. However, 12 is not prime, so it also factorizes into 3 * 4. Once again though, 4 is not prime, so it factorizes into 2 * 2.

Note: There are a few conventions when your prime factorize that you want to keep in mind:

• Order the numbers from smallest to largest

• All same numbers can be compressed into exponents

Least Common Multiple

A common multiple of two or more positive whole numbers is a number that's divisible by both numbers. For example, 24 is a common multiple of 4 and 6 because they both divide evenly into 24.

A least common multiple (LCM) is the smallest multiple that two or more numbers divide evenly into. In the example above, the least common multiple of 4 and 6 would be 12.

Naive approach to finding LCM

A simple approach to finding the LCM for especially small numbers would be to

1. Take the larger number and create a bunch of multiples

2. Check if the smaller number shares any of its multiples

Prime factorization approach to finding LCM

For larger numbers, the way to find LCM is this prime factorization approach:

1. Prime factorize the numbers in question

2. Catalog the highest occurrence of each prime factor that appears

3. Form a new product of primes using those highest occurrences and then evaluate

For example, here's this approach applied to 20 and 75:

1. Prime factorize

   • The product of primes of 20 is 2^2 * 5

   • The product of primes of 75 is 3 * 5^2

2. Catalog highest occurrences

   • The highest occurrence of prime factor 2 is 2^2

   • The highest occurrence of prime factor 3 is 3

   • The highest occurrence of prime factor 5 is 5^2

3. Form a new product of primes and evaluate

   • Therefore, the LCM of 20 and 75 is 2^2 * 3 * 5^2 = 300

Pro tip: You can double check your work by checking if the quotients of 300 / 20 and 300 / 75 share common factors other than 1. 300 / 20 = 15 and 300 / 75 = 4, and 15 and 4 share no common factors other than 1, so 300 really is the LCM of 20 and 75.

• (This seems to work because the LCM should be low enough to both numbers that dividing both numbers into the LCM shouldn't lead to very large quotients that are further divisible.)

Note: Finding the least common multiple for algebraic expressions (expressions with variables) is the same; you just treat the variables as prime and irreducible.

• 15(a^2)b and 10a(b^3)

• Reduces to 3 * 5 * a^2 * b and 2 * 5 * a * b^3

• Taking highest occurrences, that means LCM is 2 * 3 * 5 * a^2 * b^3

Greatest Common Factor

A common factor is a number that divides evenly into two or more positive whole numbers. For example, 2 divides evenly into 16 and 20, so it's a common factor.

A greatest common factor (GCF) is the largest number that divides evenly. In the example above, the GCF of 16 and 20 is 4.

Prime factorization approach to finding GCF

Just like finding the least common multiple, we do everything the same to find the greatest common factor--except we find the highest shared occurrences. For example, here's how we find the GCF for 48 and 60.

1. Prime factorize

   • The product of primes of 20 is 2^4 * 3

   • The product of primes of 60 is 2^2 * 3 * 5

2. Catalog lowest occurrences

   • The shared occurrence of prime factor 2 is 2^2

   • The lowest occurrence of prime factor 3 is 3

   • The lowest occurrence of prime factor 5 is none (b/c 20 doesn't have it)

3. Form a new product of primes and evaluate

   • Therefore, the GCF of 48 and 60 is 2^2 * 3 = 12

Pro tip: The same double check for least common multiples works with greatest common factors: since 48 / 12 = 4 and 60 / 12 = 5, 4 and 5 share no common factors other than 1, so we know 12 really is the GCF of 48 and 60.

• (This seems to work because the GCF should be high enough to both numbers that dividing into them shouldn't lead to very large quotients that are further divisible.)