Factors, Multiples, Prime Factors

Understanding Factors, Multiples and Prime Factors

A factor of a number is a whole number that can be divided evenly into that number. For instance, 1, 2, and 4 are factors of 8 because they divide 8 without leaving a remainder.
A multiple of a number is found by multiplying that number by any whole number. For example, multiples of 4 include 4 (4x1), 8 (4x2), 12 (4x3), and so on.
A prime number is a number greater than 1 that does not have any divisors other than 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, and so on.
The prime factors of a number are the prime numbers that multiply together to give that number. For example, the prime factors of 18 are 2 and 3, because 2 x 3 x 3 = 18.

To find the factors of a number, start with 1 and the number itself, as these are factors of every number. Then, divide the number by all integers up to its square root. If the division results in a whole number, both the divisor and the result are factors.
To determine the multiples of a number, simply multiply the number by integers. The resulting products are all multiples of the original number.
Beware of common errors such as confusing factors and multiples. Remember factors are always less than or equal to the number, while multiples are equal to or larger than the number.

To find the prime factors of a number, begin by dividing the number by the smallest prime number (2). If the result is a whole number, then 2 is a prime factor. Continue dividing the result by 2 until it no longer gives a whole number.
If the result isn’t divisible by 2, try dividing by the next prime number (3), and so on, until the result can’t be divided any further.
Refrain from common errors, such as stopping the process too soon. The process should be continued until the result is a prime number.
Prime factorisation is often expressed using exponential notation. For instance, the prime factorisation of 18 may be written as 2 * 3^2, which is the equivalent of 2 * 3 * 3.

Converting a number into prime factor decomposition means expressing it as a product of its prime factors. This can be achieved by systematically dividing the number by prime numbers until only prime numbers are left.
To illustrate this, the prime factor decomposition of 60 is 2^2 * 3 * 5, which is derived from progressively dividing 60 by 2 (to get 30), then 2 (to get 15), then 3 (to get 5), finishing when only the prime number 5 is left.
Take note not to mix up the order of operations. Multiplication (or division) should be executed before raising to a power in exponential notation.