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Study Guides > Prealgebra

Introduction to Prime Factorization and the Least Common Multiple

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.

Learning Outcomes

By the end of this section, you will be able to:
  • Find the prime factorization of a composite number
  • Find the least common multiple (LCM) of two numbers
 

readiness quiz

Before you get started, take this readiness quiz. #145411 Is [latex]127[/latex] prime or composite? If you missed this problem, review the following video.   Write [latex]2\cdot 2\cdot 2\cdot 2[/latex] in exponential notation.If you missed this problem, review the following examples.
Identify each number as prime or composite:
  1. [latex]83[/latex]
  2. [latex]77[/latex]

Answer: Solution: 1. Test each prime, in order, to see if it is a factor of [latex]83[/latex] , starting with [latex]2[/latex], as shown. We will stop when the quotient is smaller than the divisor.

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.
We can stop when we get to [latex]11[/latex] because the quotient [latex]\text{(7.545}\ldots\text{)}[/latex] is less than the divisor. We did not find any prime numbers that are factors of [latex]83[/latex], so we know [latex]83[/latex] is prime. 2. Test each prime, in order, to see if it is a factor of [latex]77[/latex].
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.
Since [latex]77[/latex] is divisible by [latex]7[/latex], we know it is not a prime number. It is composite.

Licenses & Attributions

CC licensed content, Shared previously

  • Ex 1: Apply Divisibility Rules to a 4 Digit Number. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Train at a station. Authored by: harlock81. Located at: https://www.flickr.com/photos/harlock81/2470743749/. License: CC BY-SA: Attribution-ShareAlike.
  • Question ID: 145433, 145411. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.

CC licensed content, Specific attribution