**High School Number Sense Lesson 85: Adding the Reciprocals of the Positive Integral Divisors of a Number**

This concept is closely related to two others we have already covered:

**NumPosIntDiv**and

**AddPosIntDiv**. It appeared

**6 times**on high school tests this year, with a median placement at

**question # 48**.

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These problems are not quite as difficult as they look. Usually, the number in question can be easily factored to find its positive integral divisors. Also, the number in question does not usually have many integral divisors. Therefore, I usually do not use the formula to add the divisors--I instead identify them and add as I go.

**How to Solve:**

- Your solution's numerator will be the sum of the positive integral divisors. Find this sum by using the formula (
**AddPosIntDiv**) or identifying them and adding as you go. - Your denominator will be the number in question.
- Reduce if necessary.

**Example 1: The sum of the reciprocals of all the positive divisors of 6 is ___**

- 6 has the positive divisors of 1, 2, 3, and 6. Their sum is
**12**; this is your numerator. **6**is your denominator.- 12/6 =
**2**.

__:__

**Note***This one is easy to test: 1/1 + 1/2 + 1/3 + 1/6 = 1 + 3/6 + 2/6 + 1/6 = 1 + 6/6 = 1 + 1 =*

**2**.**Example 2: The sum of the reciprocals of all the positive divisors of 8 is ___**

- 8 has divisors of 1, 2, 4, and 8. Their sum is
**15**; this is your numerator. **8**is your denominator.- Your answer is
**15/8**, which does not reduce.*(Do not waste your time converting this to a mixed number or decimal, unless the test requires that type of answer).*

**Example 3: The sum of the reciprocals of all the positive integral divisors of 15 is ___**

- 15 has divisors of 1, 3, 5, and 15. Their sum is
**24**, which is your numerator. **15**is your denominator.- 24/15 reduces to
**8/5**.

**Example 4: The sum of the reciprocals of all the positive integral divisors of 33 is ___**

- 33 has divisors of 1, 3, 11, and 33. Their sum is
**48**, which is your numerator. **33**is your denominator.- 48/33 reduces to
**16/11**.

**Example 5: The sum of the reciprocals of all the positive integral divisors of 18 is ___**

**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**. This is your numerator.

4.

**18**is your denominator.

5. 39/18 reduces to

**13/6**.

**Up Next for High School: CubesSolid**