**High School Number Sense Lesson 28: Finding the Number of Positive Integral Divisors**

This concept showed up

**13 times**this year--most often before question # 30, at the median spot of

**question # 27**. These problems can be solved in at least 3 ways, but I will show you only the one that is the quickest for numbers with many divisors.

**Number Dojo Level: 182**

I am going to break down this concept a little bit because it sounds more complicated than it really is:

**Positive**: greater than zero**Integral**: being an integer (whole number)**Divisor**: factor (divides evenly into the given number)

**positive integral divisors**a number has, we are looking for the total unique positive whole number factors of the number.

**How to Solve:**

**Prime factor**the number.

2. Think of the prime factoring in

**exponent notation**, for example:

*x*.

^{a}y^{b}z^{c}**Add 1**to each exponent.

4.

**Find the product**of these (augmented) exponents:

**(a + 1)(b + 1)(c + 1)**

This is easier to see with a few examples, so here we go:

**Example 1: The number of positive integral divisors of 64 is ___**

**64 = 2 x 2 x 2 x 2 x 2 x 2**

**2**

^{6}**7**

**7**.

**Example 2: 45 has ___ positive integral divisors.**

**45 = 3 x 3 x 5**

**3**

^{2}x 5^{1}**3**, and 1 + 1 =

**2**

**6**.

**Example 3: The number of positive integral divisors of 48 is ___**

**48 = 2 x 2 x 2 x 2 x 3**

**2**

^{4}x 3^{1}**5**, and 1 + 1 =

**2**

**10**.

**Example 4: 60 has ___ positive integral divisors.**

**60 = 2 x 2 x 3 x 5**

**2**

^{2}x 3^{1}x 5^{1}**3**, 1 + 1 =

**2**, and 1 + 1 =

**2**

**12**.

**Example 5: 24 has how many integral divisors? ___**

**24 = 2 x 2 x 2 x 3**

**2**

^{3}x 3^{1}**4**, and 1 + 1 =

**2**

**8**.

**Be careful!**They didn't ask for the # of

**positive**integral divisors, so we have to assume they wanted negative divisors too. Simply

**double**your result from step # 4. The answer is

**16**.

(They happen to be -24, -12, -8, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 8, 12, and 24).

**Here's a free worksheet to help you practice NumPosIntDiv:**

numposintdiv.pdf |

**Up Next for High School: MultMixDiff**