Factored Integers

A memorization exercise

Last week I added a new page to my tutoring web site. The new page’s title is “Factored Integers.” It is a reference page listing about 200 positive integers, completely factored out as products of smaller numbers, including their prime factorization.
The purpose of such a list is for some standardized test takers to do a memorization exercise. The idea is for the student to copy this list, and to write by hand a portion of it every day, anywhere from 20 to 50 numbers a day.
Just writing down the numbers and their factorizations has a cumulative effect in the student’s memory, as long as they do the exercise every day. Standardized test have many problems that can be solved much faster by factoring numbers out than by doing long multiplications and divisions.
Time is the most precious resource in a timed test, so the goal is for the student to have readily available, fresh in their memory, these factorizations, instead of wasting time thinking about what could be a possible factorization, or even worse, going down the path of long multiplications and divisions, because these operations become very time consuming, and prone to errors when the numbers involved are large.
So the best way to solve these problems is by factoring all numbers as much as possible, and simplifying all expressions as much as possible by canceling out any common factors that can be canceled out before getting into any multiplication.
So the value of the memorization exercise resides in the increased awareness of factors the student develops a little bit each day by writing and re-writing the list of factorizations.
The goal is for the student to start thinking about a number’s factors as soon as they see the number in the problem; naturally, automatically, by default, without even thinking about it. See a number, boom! Factor it. The less time you spend on this process at test time, the better. So the time spent at home writing and re-writing the list will pay off on test day.