**High School Number Sense Lesson 60: Multiplying in Bases other than Base 10**

And now, back to the bases! Last week we

**divided**in bases other than base 10. Now we will do the opposite and multiply in other bases. This concept appeared

**7 times**last year, with a median placement at

**question # 67**.

**Please Like our Facebook Page (**

**https://www.facebook.com/numberdojo/**

**) if you want to see new posts on your wall. I will reward you with a free concept index or flashcard file of your choice. Thank you!**

**Number Dojo Level: 264**

When we multiply

**in base 10**, we start with the units digit and work our way to the left, regrouping (if needed) as we go. Multiplying

**in bases other than base 10**is no different.

**How to Solve:**

- Multiply the units digit of the smaller factor by the units digit of the larger factor. If the product is less than the base, write it down.
- If the product is greater than (or equal to) the base, divide the product by the base and write down the remainder. Regroup the quotient to the next step.
- Multiply the units digit of the smaller factor by the "bases" digit of the larger factor. Add any regrouping from Step 2. If the result is less than the base, write it down.
- If the result is greater than (or equal to) the base, divide the product by the base and write down the remainder. Regroup the quotient to the next step.
- Continue this pattern until you finish multiplying each digit.

**You can also solve these by converting each factor to base 10, multiplying them together, and then converting the product back to the original base. I only recommend this method if both factors have more than one digit. However, multiplying two 2-digit numbers can be done by**

__Note__:**FOIL**ing as well.

Let's take a look at some examples.

**Example 1: 34 (base 6) x 2 (base 6) = ___ (base 6)**

- 2 x 4 =
**8**, which is greater than the base (6). Divide 8 by 6 to get**1****remainder 2**. Write down the**2**and regroup the**1**. - 2 x 3 =
**6**, plus the regrouped 1 is**7**, which is greater than the base. Divide 7 by 6 to get**1 remainder 1**. Write down**11**in front of the**2**. - Your answer is
**112**.

**Example 2: 23 (base 5) x 3 (base 5) = ___ (base 5)**

- 3 x 3 =
**9**, which is greater than the base (5). Divide 9 by 5 to get**1 remainder 4**. Write down the**4**and regroup the**1**. - 3 x 2 =
**6**, plus the regrouped 1 is**7**, which is greater than the base. Divide 7 by 5 to get**1 remainder 2**. Write down**12**in front of the**4**. - Your answer is
**124**.

**Example 3: 45 (base 7) x 6 (base 7) = ___ (base 7)**

- 6 x 5 =
**30**, which is greater than the base (7). Divide 30 by 7 to get**4 remainder 2**. Write down the**2**and regroup the**4**. - 6 x 4 =
**24**, plus the regrouped 4 is**28**, which is greater than the base. Divide 28 by 7 to get**4 remainder 0**. Write down**40**in front of the**2**. - Your answer is
**402**.

**Example 4: 312 (base 4) x 3 (base 4) = ___ (base 4)**

- 3 x 2 =
**6**, which is greater than the base (4). Divide 6 by 4 to get**1 remainder 2**. Write down the**2**and regroup the**1**. - 3 x 1 =
**3**, plus the regrouped 1 is**4**, which is equal to the base. Divide 4 by 4 to get**1 remainder 0**. Write down the**0**in front of the**2**and regroup the**1**. - 3 x 3 =
**9**, plus the regrouped 1 is**10**, which is greater than the base. Divide 10 by 4 to get**2 remainder 2**. Write down**22**in front. - Your answer is
**2202**.

**Example 5: 2016 (base 8) x 7 (base 8) = ___ (base 8)**

- 7 x 6 =
**42**, which is greater than the base (8). Divide 42 by 8 to get**5 remainder 2**. Write down the**2**and regroup the**5**. - 7 x 1 =
**7**, plus the regrouped 5 is**12**, which is greater than the base. Divide 12 by 8 to get**1 remainder 4**. Write the**4**in front of the**2**and regroup the**1**. - 7 x 0 =
**0**, plus the regrouped 1 is**1**. Write this down in front. - 7 x 2 =
**14**, which is greater than the base. Divide 14 by 8 to get**1 remainder 6**. Write down**16**in front. - Your answer is
**16142**.

**Example 6: 11 (base 2) x 22 (base 3) = ___ (base 4)**

- This is a bonus question--notice the 3 different bases. The quickest way I personally can solve this is by converting each number to base 10, multiplying, and then converting back.
- 11 (base 2) =
**3**in base 10. - 22 (base 3) =
**8**in base 10. - 3 x 8 =
**24**. 24 (base 10) =**120**in base 4. - Your answer is
**120**.

**Example 7: 23 (base 9) x 23 (base 9) = ___ (base 9)**

- Notice that each factor has 2 digits (we're squaring 23 in base 9). Convert 23 to base 10.
- 23 (base 9) =
**21**in base 10. - 21 x 21 =
**441**. To convert 441 back to base 9, divide 441 by 9. 441 ÷ 9 =**49 remainder 0**. Write down the**0**. - 49 ÷ 9 =
**5 remainder 4**. Write the**54**in front. - Your answer is
**540**.

**Example 7 (FOIL method): 23 (base 9) x 23 (base 9) = ___ (base 9)**

- 3 x 3 =
**9**, which is equal to the base. Divide 9 by 9 to get**1 remainder 0**. Write down the**0**and regroup the**1**. - 3 x 2 =
**6**, plus 2 x 3 (**6**), plus the regrouped 1 is**13**, which is greater than the base. 13 ÷ 9 =**1 remainder 4**. Write down the**4**in front and regroup the**1**. - 2 x 2 =
**4**, plus the regrouped 1 =**5**. Write this in front. - Your answer is
**540**--which matches the previous example. We must have done something right!

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

basemult.pdf |

**Up Next for High School: Logarithm**