**Middle School Number Sense Lesson 83: Finding the Remainder when Dividing by 11**

This lesson is long overdue--I have been wanting to address it for months now. It shows up so frequently that many of my students ask if there is a "better way" than using long division. Well, there is. This concept has appeared

**5 times**so far this year, with a median placement at

**question # 27**.

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This concept is related to

**RemainDiv9**, but it is a little trickier. To find the remainder when dividing by 11, instead of adding all the digits in the dividend, we need to add

*and*subtract.

**How to Solve:**

- Beginning at the right (with the units digit), add
**every other**digit in the dividend. - Subtract each digit skipped in Step 1.
- If the result is a non-negative integer less than 11, you have your answer. If not, add or subtract 11 to get it in that range.

**Example 1: 8723 ÷ 11 has a remainder of ___**

- Beginning at the right, add every other digit: 3 + 7 =
**10**. - Subtract each skipped digit: 10 - 8 - 2 =
**0**. - The answer is
**0**.

**Example 2: 72683 ÷ 11 has a remainder of ___**

- Beginning at the right, add every other digit: 3 + 6 + 7 =
**16**. - Subtract each skipped digit: 16 - 2 - 8 =
**6**. - The answer is
**6**.

**Example 3: 63947 ÷ 11 has a remainder of ___**

- Beginning at the right, add every other digit: 7 + 9 + 6 =
**22**. - Subtract each skipped digit: 22 - 3 - 4 =
**15**. - Since 15 > 10, subtract 11. 15 - 11 =
**4**, which is your answer.

**Example 4: 812365 ÷ 11 has a remainder of ___**

- Beginning at the right, add every other digit: 5 + 3 + 1 =
**9**. - Subtract each skipped digit: 9 - 8 - 2 - 6 =
**-7**. - Since -7 < 0, add 11. -7 + 11 =
**4**, which is your answer.

**Example 5: 4574730 ÷ 11 has a remainder of ___**

- Beginning at the right, add every other digit: 0 + 7 + 7 + 4 =
**18**. - Subtract each skipped digit: 18 - 5 - 4 - 3 =
**6**. - The answer is
**2**.

**Example 6: 319232017 ÷ 11 has a remainder of ___**

- Beginning at the right, add every other digit: 7 + 0 + 3 + 9 + 3 =
**22**. - Subtract each skipped digit: 22 - 1 - 2 - 2 - 1 =
**16**. - Since 16 > 10, subtract 11. 16 - 11 =
**5**, which is your answer.

**Up Next for Middle School: AddPosIntDiv**