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أدلة الدراسة > 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 1515 minutes, one arrives every 1212 minutes, and one arrives every 99 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 127127 prime or composite? If you missed this problem, review the following video.   Write 22222\cdot 2\cdot 2\cdot 2 in exponential notation.If you missed this problem, review the following examples.
Identify each number as prime or composite:
  1. 8383
  2. 7777

Answer: Solution: 1. Test each prime, in order, to see if it is a factor of 8383 , starting with 22, as shown. We will stop when the quotient is smaller than the divisor.

Prime Test Factor of 83?83?
22 Last digit of 8383 is not 0,2,4,6,or80,2,4,6,\text{or}8. No.
33 8+3=118+3=11, and 1111 is not divisible by 33. No.
55 The last digit of 8383 is not 55 or 00. No.
77 83÷7=11.85783\div 7=11.857\ldots No.
1111 83÷11=7.54583\div 11=7.545\ldots No.
We can stop when we get to 1111 because the quotient (7.545)\text{(7.545}\ldots\text{)} is less than the divisor. We did not find any prime numbers that are factors of 8383, so we know 8383 is prime. 2. Test each prime, in order, to see if it is a factor of 7777.
Prime Test Factor of 77?77?
22 Last digit is not 0,2,4,6,or 80,2,4,6,\text{or }8. No.
33 7+7=147+7=14, and 1414 is not divisible by 33. No.
55 the last digit is not 55 or 00. No.
77 77÷11=777\div 11=7 Yes.
Since 7777 is divisible by 77, we know it is not a prime number. It is composite.

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