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Study Guides > ALGEBRA / TRIG I

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

readiness quiz

1. If you missed this problem, review the following example.
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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