Factors vs Multiples
Factors and multiples are two different but related topics in Basic Algebra. Factors and multiples lead towards the lesson of factoring. The concept of factoring is very simple, but an important topic as it has wide ranging application in real world.
Factor
In Mathematics, a Factor, also called a divisor is an integer or algebraic expression that divides another number or expression without leaving a reminder. Factor can be positive as well as negative. This includes 1 and the number itself. For example, 2 is a factor of 14 because 14/2 is exactly 7. The factors of 14 are 1, 2, 7, 14, 1, 2,7 and 14 (but only the positive ones would usually be mentioned, i.e. 1, 2, and 4.). For another example, x+3 is a factor of the algebraic expression x^{2}+11x+24.
A positive integer greater than 1 or an algebraic expression that has only two factors, 1 and the number itself is termed prime. For example 5 is a prime number, since it is divisible only by 1 and the number itself. On the other hand, if a positive integer or an algebraic expression has more than two factors, it is called composite. For example, 6 is evenly divisible by both 2 and 3, in addition to 1 and itself. Since number 1 has exactly one factor ‘1’, it is neither prime nor composite. We can write any number as a product of its factors. For example, we can write 12 as a product of 2 and 6 (i.e. 12=2×6) and also as product of 3 and 4 (i.e. 12=3×4).
Multiple
A multiple of a number is the result of multiplying that number by any other whole number. Multiples, on the other hand, are the products of factors. For the quantities a and b we say that a is a multiple of b, if a = nb for some integer of n, where n is called the multiplier. For example, 5, 10, 15 are multiples of 5 because these numbers can be written as a product of 5 and another integer. 0 is a multiple of any number and every number is a multiple itself.
What is the difference between Factors and Multiples?  Factors are made up of multiplicand and multiplier, or divisor and dividend; while Multiples are the product of factors.  Multiples, on the other hand, are the products of factors.
