1.4.3 Finding a Remainder when Dividing by 11

Finding the remainder when dividing by 11 is very similar to finding the remainder when dividing by 9 with one catch: you add up alternating digits (beginning with the ones digits) then subtract the sum of the remaining digits. Let’s look at an example to illustrate the trick:

13542 ÷ 11 has what remainder?

Sum of the Alternating Digits: $(2 + 5 + 1) = 8$
Sum of the Remaining Digits: $(4 + 3) = 7$
Remainder: $8 - 7 = 1$

Sometimes adding then subtracting “down the digits” will be easier than finding two explicit sums then subtracting. For example, if we were finding the remainder of $3456789 \div 11$ , instead of doing $(9 + 7 + 5 + 3) - (8 + 6 + 4) = 24 - 18 = 6$ it might be easier to do $9 - 8 + 7 - 6 + 5 - 4 + 3 = 1 + 1 + 1 + 3 = 6$ . That is what is so great about number sense tricks, is there are always methods and approaches to making them faster!

Problem Set 1.4.3

7653 \div 11 \text{ remainder}

745321 \div 11 \text{ remainder}

142536 \div 11 \text{ remainder}

6253718 \div 11 \text{ remainder}

87125643 \div 11 \text{ remainder}

325476 \div 11 \text{ remainder}

\text{Find } k \text{ so } 23578k \text{ is divisible by 11}

\text{Find } k \text{ so } 1482065k5 \text{ is divisible by 11}

\text{Find } k \text{ so } 456k89 \text{ is divisible by 11}

\text{Find } k \text{ so } 377337k \text{ is divisible by 11}