# Factors

## … And The Oldest Has Red Hair

It’s been a while since we’ve done a puzzle, just for fun. Here we’ll look at two versions of a riddle, about finding children’s ages from a known product, a partially known sum, and a bizarre fact about the oldest. Then we’ll close with an interesting variation.

## Prime Factorization of a Number (Advanced)

Last time we looked at basic methods for finding the prime factorization of a number. Here we will look at some special techniques for large numbers, demonstrating them for not-too-large numbers. This takes us a step beyond previous tests that told us whether a number was composite, without actually factoring them.

## Partial Fractions: Complex and Trivial Cases

(A new question of the week) We discussed four years ago how to make a partial fraction decomposition of a rational function, and why it can always be done; a question from mid-May brings up two side issues: when you can factor the denominator, and whether a trivial decomposition, which takes no work at all, …

## Two Word Problems About Factors and Sums

(A new question of the week) A couple recent questions involved factoring numbers, in interesting ways. One involves the volume and perimeter of a block of cubes, and the other involves finding numbers with a given HCF (Highest Common Factor) and sum. Both illustrate thinking through a non-routine problem about factors.

## Help with Factoring: Trinomials

(A new question of the week) I recently had a pleasant discussion of factoring, with the kind of student for whom I returned to teaching: one who has been away from math for a while, and with greater maturity has the determination to succeed. We’ll see several examples of the “ac-grouping” method of factoring a …

## A Limit: Getting the Algebra Right

(A new question of the week) I have often said that calculus class is where many students finally learn algebra, because now algebra is an essential tool, not just something to learn for an exam. This is especially true of a nontraditional student, who may not have taken math recently, or may even be learning …

## Summing Divisors

In searching for answers about counting divisors over the last couple weeks, I found a few that are about the similar question of finding the sum of a number’s divisors. In fact, a couple questions and answers confuse the two problems. Let’s finish off the topic by looking at these. (Keep in mind that “divisor” …

## The Locker Problem

A classic problem we’ve seen hundreds of times involves students opening and closing lockers. I have often told people that, believe it or not, they could find the answer by searching the Ask Dr. Math site for the word “locker”. But I prefer to give them a reference to one of the answers in which …

## One More Way to Find GCF and LCM

There are so many ways to find a Least Common Multiple that I had to omit one method we have been asked about several times. This one doesn’t require finding prime factors, but focuses on division by whatever factors you see. Divide everything by whatever works The first reference to the method I have found …

## Many Ways to Find the Least Common Multiple

Last time, we looked at three ways to find a GCF (Greatest Common Factor). Here we’ll see the corresponding ways to find an LCM (Least Common Multiple); next time we’ll examine another method, which in its full form finds both at once. (Note that the Least Common Denominator of fractions is just the Least Common …