**High School Number Sense Lesson 62: Relatively Prime**

A concept that has been showing up a lot recently is the idea of

**relative primeness**. Last year it appeared

**11 times** on high school tests, with a median placement at

**question # 64**.

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**Definition:**

Two integers are relatively prime (or

**coprime**) if the only positive integer that divides evenly into both integers is

**1**. In other words, their

**Greatest Common Factor**is 1.

The most common type of problem in this category looks something like this:

**How many integers less than or equal to 20 are relatively prime to 20?**

**How to Solve:**

- Prime factor the number. (See
**PrimeFactor**if you need a refresher course). In this case, the prime factorization of**20**is**2 x 2 x 5**. - For each
**unique**prime factor, create a fraction that has the**prime as the denominator**, and**the prime minus 1 as the numerator**. So for this example,**1/2**and**4/5**.**Do not duplicate the 1/2 even though 2 is a factor twice**. - Multiply the original number by each of these fractions: 20 x 1/2 =
**10**. 10 x 4/5 =**8**. Your answer is**8**.

**Why it Works:**

- Think of all the integers less than or equal to 20 (in groups of 5):

**1 2 3 4 5**

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

- Now eliminate every
**even integer**, because 20 cannot be relatively prime to any even integer. We're left with:

**1 3 5**

7 9

11 13 15

17 19

7 9

11 13 15

17 19

- Now eliminate every
**multiple of 5**. We're left with:

**1 3**

7 9

11 13

17 19

7 9

11 13

17 19

- We have
**8**integers left--which is what we got in our previous calculation. :)

Example 1: How many integers less than 14 are relatively prime to 14?

Example 1: How many integers less than 14 are relatively prime to 14?

- Prime factor 14:
**2 x 7**. - Create your fractions with 2 & 7 as denominators:
**1/2**and**6/7**. - Multiply the original number by both fractions. 14 x 1/2 =
**7**. 7 x 6/7 =**6**, which is your answer.

**Example 2: How many integers less than 44 are relatively prime to 44?**

- Prime factor 44:
**2 x 2 x 11**. - Create your fractions with 2 & 11 as denominators:
**1/2**and**10/11**. - Multiply the original number by both fractions. 44 x 1/2 =
**22**. 22 x 10/11 =**20**, which is your answer.

**Example 3: How many integers less than 45 are relatively prime to 45?**

- Prime factor 45:
**3 x 3 x 5**. - Create your fractions with 3 & 5 as denominators:
**2/3**and**4/5**. - Multiply the original number by both fractions. 45 x 2/3 =
**30**. 30 x 4/5 =**24**, which is your answer.

**Example 4: How many integers less than 30 are relatively prime to 30?**

- Prime factor 30:
**2 x 3 x 5**. - Create your fractions with 2, 3 & 5 as denominators:
**1/2**,**2/3**, and**4/5**. - Multiply the original number by both fractions. 30 x 1/2 =
**15**. 15 x 2/3 =**10**. 10 x 4/5 =**8**, which is your answer.

**Example 5: How many integers less than 20 are relatively prime to 14?**

- Be careful here: we're looking at 2 different numbers (20 and 14). Prime factor 14:
**2 x 7**. - Create your fractions with 2 & 7 as denominators:
**1/2**and**6/7**. - Multiply the original number by both fractions. 14 x 1/2 =
**7**. 7 x 6/7 =**6**. - Now think of the integers between 14 & 20 that are also relatively prime to 14:
**15, 17, and 19**. There are**3**of them, so add 3 to the 6 and get**9**, which is the answer.

**Example 6: How many integers between 2 and 20 are relatively prime to 20?**

- Be careful here: we're given the range of
**2 - 20**. Prime factor 20:**2 x 2 x 5**. - Create your fractions with 2 & 5 as denominators:
**1/2**and**4/5**. - Multiply the original number by both fractions. 20 x 1/2 =
**10**. 10 x 4/5 =**8**. - Now think of our range of 2 - 20. This excludes 2 (which we've already eliminated) and 1 (which we haven't)... So subtract 1 from 8 to get
**7**, which is the answer.

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

primerelative.pdf |

**Up Next for High School: RepDecFracDec**