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Your aunt has an excellent recipe for a breakfast sandwich that her customers love, but the cafe is all out of eggs and buns. She needs you to run to the grocery store for supplies. Exactly one egg and one bun are required to make each sandwich. Unfortunately, you can only purchase eggs in packages of 12 and buns in packages of eight. If you return with one dozen eggs and one eight-package of buns, four eggs will go unused. If you return with one dozen eggs and two packages of buns, four buns will be left over. What is the smallest number of eggs and buns you can purchase so there are no leftovers of eggs or buns? (Feel free to take out a sheet of paper and work the numbers! It will not take long to see that the answer is 24).
In this brief unit, we explore the concepts of least common multiple (LCM) and greatest common factor (GCF). Our run-of-the-mill questions, such as our egg-and-bun problem, indicate how these concepts are relevant to our everyday lives, but they also show up in lots of other mathematical questions. Here we are talking about sets of whole numbers: in Unit 1, our operations applied to whole numbers, fractions, and every other type of real number. Consequently, notions of least common multiple and greatest common factor belong to the part of mathematics that studies patterns among whole numbers, an area called number theory.
Completing this unit should take you approximately 1 hour.
Before we discuss great or common factors, let's use an example to help define what a factor is all by itself. Take the number 12. We can write , which shows that three divides into 12 with zero remainder (and that four divides into 12 too). We then say that three is a factor of 12 (and that four is a factor of 12 too). We can also write , so two and six are also factors of 12. We can also write so and are also factors of 12. In fact, all together, we have:
The (positive) factors of 12 are , and
More generally, an integer is a factor of a number means that . Some numbers have lots of factors, while others are special because they have just a few factors. For example, the only (positive) factors of the number 11 are one and itself. We call a number that has exactly two factors, such as 11, a prime number.Number | Positive Factors | ||||
12 | 1 | 2 | 3 | 4 | 12 |
8 | 1 | 2 | 4 | 8 |
Here's another example. Consider the numbers 50, 75, and 100. Can you work out that GCF(50, 75, 100) = 25?
Watch this video for examples showing how to determine GCF for a set of numbers. Note that sometimes GCF is called the greatest common divisor.
Read this text. Pay close attention to the section, "A Method for Determining the Greatest Common Factor". Complete the practice questions and check your answers.
Multiples and factors are related concepts. Instead of saying three is a factor of 12, we could write 12 is a multiple of 3. In general, we say that one number is a multiple of another number when (another integer).
We can arrange the (positive) multiples of three, for example, in a list like this one:
Positive Multiples | |||||||
Number | 1 | 2 | 3 | 4 | 5 | 6 | ... |
12 | 12 | 24 | 36 | 48 | 60 | 72 | ... |
8 | 8 | 16 | 24 | 32 | 40 | 48 | ... |
Read this text for a description of multiples and the method to determine the least common multiple of a given set of numbers. Complete the practice questions and check your answers.
Watch this video for more examples of determining the least common multiple for a set of three numbers.
Finally, watch this video to see applied examples of using the least common multiple and greatest common factor.