Divisibility rules, prime numbers and prime factorization

Factors and multiples

\(6 \cdot 7 = 42\), so \(6\) and \(7\) are factors of \(42\), and \(42\) is a multiple of both \(6\) and \(7\). Another way to say it: \(42\) is divisible by \(6\), because \(42 \div 6\) leaves no remainder.

• The multiples of a number go on forever: the multiples of \(8\) are \(8, 16, 24, 32, \dots\)
• The factors of a number are limited. To find them all, look for factor pairs, starting from \(1\) and stopping when the pairs start repeating.

Divisibility rules

You can often tell whether a number is divisible by another without doing the division.

Prime and composite numbers

The primes less than \(40\) are

Two facts trip people up:

• \(1\) is neither prime nor composite. It has only one factor.
• \(2\) is the only even prime. Every other even number has \(2\) as a factor.

To test whether a number is prime, try dividing by \(2, 3, 5, 7, 11, \dots\) You can stop once the prime you're testing, squared, is bigger than the number. For example, \(89\) isn't divisible by \(2\), \(3\), \(5\) or \(7\), and \(11^2 = 121 > 89\), so \(89\) is prime.

Prime factorization

Every composite number can be written as a product of primes in exactly one way, apart from the order. That product is its prime factorization.

A factor tree is a neat way to find it. Split the number into any two factors, then keep splitting until every branch ends in a prime.

Prime factorizations make the next lesson, on GCF and LCM, much faster.

Practice