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Aliquot Part: Definition & Examples

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will look at the definition of an aliquot part of a number. We will explore how it relates to division and is actually the same thing as a well-known concept within division.

A Problem of Divisors

Suppose Alice has 8 tasks that she needs to hire people to do. She puts an ad in the paper and 7 people apply for a job with her. She is trying to decide how many people she should hire, knowing that she wants to be able to dole the tasks out evenly among the hires. In other words, she wants the number of people she hires to divide into the number of tasks, 8, evenly.

Let's see here. She has the choice of hiring 1, 2, 3, 4, 5, 6, or all 7 people that applied. Looking at each of these options gives the following:

  • If she hires 1 person, she could give all 8 tasks to that person, so the tasks would be doled out evenly. Thus, this is a possibility.
  • If she hires 2 people, she could give each of them 4 tasks, so the tasks would be doled out evenly. Therefore, this is another possibility.
  • If she hires 3 people, we run into a problem. Because 3 does not divide into 8 evenly, she can't divide the 8 tasks evenly among the 3 people she hired, so this is not a possibility.
  • If she hires 4 people, she could give each of them 2 tasks, so the tasks would be doled out evenly, and this is a possibility.
  • If she hired 5, 6, or 7 people, we would run into the same problem as we did with 3 people, because none of 5, 6, or 7 divide into 8 evenly, so the tasks couldn't be divided among the hires evenly in any of these cases. None of these are possibilities.

If she hired 8 people, she could give each of them one task, so they would be divided up evenly. However, this is not an option, because only 7 people replied to her add.

All together, we have that in order to be able to dole out the 8 tasks evenly, she could hire 1, 2, or 4 people. Now, let's relate this to mathematics!

Proper Divisors and Aliquot Parts

What Alice is actually looking for is the aliquot parts of the number 8. You wee, the possible number of hires, 1, 2, and 4, all divide into 8 evenly. A number that divides into a number, n, evenly is called a divisor of n. For example, divisors of 8 are 1, 2, 4, and 8, because these all divide into 8 evenly.

We can actually classify the numbers 1, 2, and 4 even further. In math, a positive proper divisor of a number, n, is a divisor of n that is positive and is not equal to n. In other words, the positive proper divisors of a number n are all of its positive divisors except itself. Therefore, the numbers 1, 2, and 4 are the positive proper divisors of 8, because they are all of eight's positive divisors except for eight itself.

Now, all of this may be information that you are already familiar with. Plus, the lesson of the title says aliquot part, not positive proper divisors. Aliquot part sounds a bit intimidating! However, you will be pleasantly surprised to know that the aliquot parts of a number are the exact same thing as the number's positive proper divisors! That is, an aliquot part of a number, n, is a divisor of n that is positive and is not equal to n itself. Huh! Well, that's simple enough!

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What a relief to know that this is such a simple concept! Let's take a look at some examples to really familiarize ourselves with the aliquot parts of a number.

Examples

Let's find the aliquot parts of the numbers 28 and 50. To do this, we simply need to find the positive proper divisors of each of these numbers. That is, we need to find the positive divisors of these numbers excluding the numbers themselves.

Consider 28 first. First, we will list the divisors of 28 that are positive, or the positive numbers that divide into 28 evenly.

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