**Middle School Number Sense Lesson 58: Dividing in Bases other than Base 10**

Several months ago, we discussed converting either direction between base 10 and another base (in

**BaseNtoBase10**and

**Base10toBaseN**). In November, we performed addition operations in bases other than 10 (in

**BaseAdd**). Before proceeding with today's lesson, you may want to revisit those concepts to have them fresh on your mind.

Today we will divide in bases other than base 10. This concept appeared

**6 times**last year--half the time at

**question # 56**and half the time at # 69.

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There are 2 methods that I use to solve these problems:

- The "
**traditional**" way of dividing, left to right--like we learned in elementary school, except in a different base. - By
**converting**the dividend and the divisor to base 10, dividing, and then converting the quotient back to the original base.

**Example 1 (traditional): 54 (base 6) ÷ 2 (base 6) = ___ (base 6)**

- 2 goes into 5,
**2**times, with a remainder of 1 (six instead of 10). Write the**2**. - Add the remainder of 1 six to the 4 in the units place to get 10. 2 goes into 10,
**5**times. Write the**5**after the 2. Your answer is**25**.

**Example 1 (converting): 54 (base 6) ÷ 2 (base 6) = ___ (base 6)**

- Using
**BaseNtoBase10**, 54 base 6 is (6 x 5 + 4) =**34**. 2 base 6 is still**2**. - In base 10, 34 ÷ 2 =
**17**. - Using
**Base10toBaseN**, 17 base 10 is**25**in base 6.

**Example 2 (traditional): 71 (base 9) ÷ 4 (base 9) = ___ (base 9)**

- 4 goes into 7,
**1**time, with a remainder of 3 (nines). Write the**1**. - Add the remainder of 3 nines (which is
**27**) to the 1 in the units place to get 28. 4 goes into 28,**7**times. Write the**7**after the 1. Your answer is**17**.

**Example 2 (converting): 71 (base 9) ÷ 4 (base 9) = ___ (base 9)**

- Using BaseNtoBase10, 71 base 9 is (9 x 7 + 1) =
**64**. 4 base 9 is still**4**. - In base 10, 64 ÷ 4 =
**16**. - Using Base10toBaseN, 16 base 10 is
**17**base 9.

**Example 3: 43 (base 8) ÷ 7 (base 8) = ___ (base 8)**

- Let's do this one the traditional way. 7 goes into 4,
**0**times, with a remainder of 4 (eights).**Don't write the 0**. - Add the remainder of 4 eights (which is
**32**) to the 3 in the units place to get**35**. 7 goes into 35,**5**times. Write the**5**, which is your answer. - (This one may have been easier to do by converting instead).

**Example 4: 123 (base 4) ÷ 3 (base 4) = ___ (base 4)**

- Let's do this one by converting. 123 base 4 is [(4 x 1 + 2) x 4] + 3, which is
**27**. 3 base 4 is still**3**. - In base 10, 27 ÷ 3 =
**9**. - 9 base 10 is
**21**base 4.

**Example 5: 202 (base 7) ÷ 5 (base 7) = ___ (base 7)**

- Let's do this one the traditional way. 5 goes into 2,
**0**times, with a remainder of 2 (sevens).**Don't write the 0**. - Add the remainder of 2 sevens (which is
**14**) to the 0 in the sevens place to get**14**. 5 goes into 14,**2**times, with a remainder of 4 (sevens). Write the**2**. - Add the remainder of 4 sevens (which is
**28**) to the 2 in the units place to get**30**. 5 goes into 30,**6**times. Write the**6**after the 2. Your answer is**26**.

**Example 6: 211 (base 5) ÷ 4 (base 5) = ___ (base 5)**

- Let's do this one by converting. 211 base 5 is [(5 x 2 + 1) x 5] + 1, which is
**56**. 4 base 5 is still**4**. - In base 10, 56 ÷ 4 =
**14**. - 14 base 10 is
**24**base 5.

**Example 7: 2016 (base 9) ÷ 3 (base 9) = ___ (base 9)**

- Let's do this one the traditional way. 3 goes into 2,
**0**times, with a remainder of 2 (nines).**Don't write the 0**. - Add the remainder of 2 nines (which is
**18**) to the 0 in the 81's place to get**18**. 3 goes into 18,**6**times, with a remainder of 0 (nines). Write the**6**. - Add the remainder of 0 nines (which is
**0**) to the 1 in the nines place to get**1**. 3 goes into 1,**0**times, with a remainder of 1 (nine). Write the**0**after the 6. - Add the remainder of 1 nine (which is 9) to the 6 in the units place to get
**15**. 3 goes into 15,**5**times. Write the**5**at the end. Your answer is**605**.

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

basediv.pdf |

**Up Next for Middle School: EstSquare**