Lesson Fourteen: In Which We Factor

So far we have seen how you can go from 1/2 to 3/6 by multiplying both the numerator and denominator by 3. Thus 1/2 = (1/2)*1 = (1/2)*(3/3) = 3/6. But suppose we were given 3/6, how would we know that it can be written as 1/2? This would be something good to know, as 1/2 is arguably "simpler" than 3/6 (the numbers involved are smaller at least).

Factoring:

If you have an integer, m, which can be written as the product of two other integers, say a and b, so m = a*b, then we call a and b factors of m. We also say that m is a multiple of a and that m is a multiple of b. So, if we consider 3, we can think of it as having factors 1 and 3, and 6 has factors 2 and 3. Since 3 is a factor of both the numerator and the denominator, we can remove it, and obtain 1/2. To express it with numbers, 3/6 = (1*3)/(2*3) = (1/2)*(3/3) = 1/2.

We really only need to consider factoring positive integers, because a negative integer is just a positive integer multiplied by -1. So, if we want to factor a negative number, if we first factor out the -1 we can proceed as though we were factoring a positive number. For example -6 = -1*6 = -1*(2*3).

Factoring Tricks:

While figuring out the factors of a number is not always an easy task, there are some tricks that can be helpful. If a number is even, it means that 2 is a factor of the number. Even numbers can be recognized by having a 0,2,4,6, or 8 in the one's place. If the sum of the digits in a number is a multiple of 3, then the original number is a multiple of three. For example consider 33, the sum of the digits is 3+3 = 6, and 6 is a multiple of 3, as we saw above, so 33 should have 3 as a factor. In fact, 33 = 11*3. Finally, if a number has a 0 or a 5 in the one's place, then it has 5 as a factor.