Introduction to Prime Factorization and the Least Common Multiple

What you’ll learn to do: Use prime factorization to find the least common multiple of a number

A modern-looking passenger train at a station platform

When will the train arrive?

Peter is exploring a new city, and he’s getting around by train. There are three train lines that leave from the station closest to his hostel. One arrives every [latex]15[/latex] minutes, one arrives every [latex]12[/latex] minutes, and one arrives every [latex]9[/latex] minutes. If all the trains depart the station at the same time every morning, how long will it be before they’re all at the station at the same time again? To find this out, you’ll use prime factorization and find the least common multiple–we’ll explore both of those concepts in this section.

Before you get started in this module, try a few practice problems and review prior concepts.

readiness quiz

2. Is [latex]810[/latex] divisible by [latex]2,3,5,6,\text{ or }10?[/latex]

Answer: [latex]2, 3, 5, 6,\text{ and }10[/latex]

If you missed this problem, review the following video.

4.

If you missed this problem, review the following example.

Identify each number as prime or composite:

[latex]83[/latex]
[latex]77[/latex]

Show Solution

Prime	Test	Factor of [latex]83?[/latex]
[latex]2[/latex]	Last digit of [latex]83[/latex] is not [latex]0,2,4,6,\text{or}8[/latex].	No.
[latex]3[/latex]	[latex]8+3=11[/latex], and [latex]11[/latex] is not divisible by [latex]3[/latex].	No.
[latex]5[/latex]	The last digit of [latex]83[/latex] is not [latex]5[/latex] or [latex]0[/latex].	No.
[latex]7[/latex]	[latex]83\div 7=11.857\ldots[/latex]	No.
[latex]11[/latex]	[latex]83\div 11=7.545\ldots[/latex]	No.

Prime	Test	Factor of [latex]77?[/latex]
[latex]2[/latex]	Last digit is not [latex]0,2,4,6,\text{or }8[/latex].	No.
[latex]3[/latex]	[latex]7+7=14[/latex], and [latex]14[/latex] is not divisible by [latex]3[/latex].	No.
[latex]5[/latex]	the last digit is not [latex]5[/latex] or [latex]0[/latex].	No.
[latex]7[/latex]	[latex]77\div 11=7[/latex]	Yes.

Chapter 4: Simple Equations and Inequalities