Factorization, products, powers and indices Factors

Factorization, products, powers and indices Factors

When two or more numbers are multiplied together, each of them, or the product of any number of them (apart from them all), is a factor of the product. This applies to explicit arithmetic numbers and to literal numbers. So, for example, if we multiply the numbers 2 and 6, we get that 2 X 6 = 12, thus 2 and 6 are factors of the number 12. However, the number 12 has more than one set of factors, 3 X 4 = 12 so 3 and 4 are also factors of the number 12. We can also multiple 2 X 2 X 3 to get 12. So, the numbers 2, 2 and 3 are yet another set of factors of the number 12. Finally, you will remember that any number n multiplied by 1 is itself, or n X 1 = n. So, every number has itself and 1 as factors. 1 and n are considered trivial factors and when asked to find the factors of an explicit or literal number, we will exclude the number itself and 1.

Example 4.7

Find the factors of:     (a) 8,          (b) xy (c) 24, (d) abc           and     (e) –m.

(a) Apart from the trivial factors 1 and 8, which we agreed to ignore, the number 8 has the factors 2 X 2 x 2 and 2 x 4 since, 2 X 2 X 2 = 8 and 2 X 4 = 8, remember that this last set of factors can be presented in reverse order, 4 x 2 = 8, but although the factor 2 is repeated the factors 2 and 4 are still the only factors.

(b) Similarly, the literal number xy can only have the factors x and y, if we ignore the trivial factors. Thus the numbers x and y multiplied together to form the product xy are factors of that product.

(c) The number 24 has several sets of factors with varying numbers in each set. First we find the number of sets with two factors, these are:

24=6X4

24=8X3

24=12X2

More than two factors:

24 =2 X2 X6

24 =4 X3 X2

 24=2 X2 X2 X3

However if we look closely we see that the number 24 has only six different factors: 12,8,6,4,3and2.

(d) So, what about the factors in the number abc? Well I hope you can see that the product of each individual factor a, b and c constitute one set of factors. Also ab and c;a and bc; and b and ac, form a further three sets. So, extracting the different factors from these sets we have: a, b, c, ab, ac and bc as the six factors of the number abc.

(e) We have two sets of factors here 1 and –n, which are the trivial factors, but also the set n and –1, notice the subtlety with the sign change. When dealing with minus numbers, any two factors must have opposite signs.