**Middle School Number Sense Lesson 84: Finding the Sum of the Positive Integral Divisors of a Number**

This concept is an extension of

**NumPosIntDiv**, which we discussed last August. I recommend you master that concept before trying to learn this one, because many of the steps carry over. Today's concept has appeared

**7 times**so far this year, with a median placement at

**question # 36**.

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The easiest method of adding the positive integral divisors of a

**small-ish**number is to identify them from the outside in, and add as you go.

**Example 1: The sum of the positive integral divisors of 15 is ___**

- Identify the divisors from the outside in:
**1 x 15**, and**3 x 5**. - Add as you go: 1 + 15 =
**16**, and 3 + 5 =**8**, and 16 + 8 =**24**.

**Example 2: The sum of the positive integral divisors of 12 is ___**

- Identify the divisors from the outside in:
**1 x 12**,**2 x 6**, and**3 x 4**. - 1 + 12 =
**13**, 2 + 6 =**8**, and 3 + 4 =**7**. 13 + 8 + 7 =**28**.

The formula for this process takes some practice, but it is worth learning--especially for larger numbers.

**How to Solve:**

**Prime factor**the number, and think of it in exponent notation.- For each prime factor,
**add 1**to the exponent, and**subtract 1**. Then**divide**by (the prime factor minus 1). **Multiply**all the results from Step 2.

**Example 1 (again): The sum of the positive integral divisors of 15 is ___**

**15 = 3**.

^{1}x 5^{1}2. (3

^{2}- 1)/(3 - 1) = 8/2 =

**4**. (5

^{2}- 1)/(5 - 1) = 24/4 =

**6**.

3. 4 x 6 =

**24**.

**Example 2 (again): The sum of the positive integral divisors of 12 is ___**

**12 = 2**.

^{2}x 3^{1}2. (2

^{3}- 1)/(2 - 1) = 7/1 =

**7**. (3

^{2}- 1)/(3 - 1) = 8/2 =

**4**.

3. 7 x 4 =

**28**.

**Example 3: The sum of the positive integral divisors of 18 is ___**

**18 = 2**.

^{1}x 3^{2}2. (2

^{2}- 1)/(2 - 1) = 3/1 =

**3**. (3

^{3}- 1)/(3 - 1) = 26/2 =

**13**.

3. 3 x 13 =

**39**.

**Example 4: The sum of the positive integral divisors of 30 is ___**

**30 = 2**.

^{1}x 3^{1}x 5^{1}2. (2

^{2}- 1)/(2 - 1) = 3/1 =

**3**. (3

^{2}- 1)/(3 - 1) = 8/2 =

**4**. (5

^{2}- 1)/(5 - 1) = 24/4 =

**6**.

3. 3 x 4 x 6 =

**72**.

**Example 5: The sum of the positive integral divisors of 72 is ___**

**72 = 2**.

^{3}x 3^{2}2. (2

^{4}- 1)/(2 - 1) = 15/1 =

**15**. (3

^{3}- 1)/(3 - 1) = 26/2 =

**13**.

3. 15 x 13 =

**195**.

**Up Next for Middle School: SolveXSeq**