Divisibility by 3 and 9

Show your work for each problem.

1.
Which number is divisible by $3$ ?
1. a)535
2. b)770
3. c)421
4. d)612
Answer: 612
The digit sums are $7$ , $13$ , $9$ and $14$ . Only $6 + 1 + 2 = 9$ is a multiple of $3$ , so $612$ is divisible by $3$ : $612 \div 3 = 204$ .
2.
Which number is divisible by $9$ ?
1. a)6303
2. b)8120
3. c)2346
4. d)5013
Answer: 5013
The digit sums are $15$ , $11$ , $12$ and $9$ . Only $5 + 0 + 1 + 3 = 9$ is a multiple of $9$ , so $5013 \div 9 = 557$ . Note: $2346$ and $6303$ are divisible by $3$ , but not by $9$ .
3.
Which statement is true?
1. a)Every number divisible by 9 is also divisible by 3
2. b)Every number divisible by 3 is also divisible by 9
3. c)Every number ending in 3 is divisible by 3
4. d)A number is divisible by 9 if its last digit is 9
Answer: Every number divisible by 9 is also divisible by 3
Since $9 = 3 \times 3$ , every multiple of $9$ is also a multiple of $3$ . The others are false: $12$ is divisible by $3$ but not by $9$ , $13$ ends in $3$ but is not divisible by $3$ , and $19$ ends in $9$ but is not divisible by $9$ .
4.
What is the remainder when $5237$ is divided by $9$ ?
1. a)0
2. b)7
3. c)8
4. d)2
Answer: 8
$5 + 2 + 3 + 7 = 17$ and $1 + 7 = 8$ , so the remainder is $8$ . Check: $9 \times 581 = 5229$ and $5237 - 5229 = 8$ .
5.
Leo wrote $3456 + 2789 = 6235$ . What does casting out nines tell you?
1. a)The answer is wrong: both sides give remainder 0
2. b)The answer passes the check, so it is correct
3. c)The answer is wrong: the left side gives remainder 8, but 6235 gives 7
4. d)Casting out nines cannot be used for addition
Answer: The answer is wrong: the left side gives remainder 8, but 6235 gives 7
$3456 \rightarrow 18 \rightarrow 9$ (remainder $0$ ) and $2789 \rightarrow 26 \rightarrow 8$ , so the sum should leave remainder $0 + 8 = 8$ . But $6235 \rightarrow 16 \rightarrow 7$ . The check does not match, so the answer is wrong (the right sum is $6245$ ).
6.
What is the digit sum of $8457$ ?
Answer: 24
$8 + 4 + 5 + 7 = 24$ . Since $24$ is a multiple of $3$ but not of $9$ , $8457$ is divisible by $3$ but not by $9$ .
7.
Add the digits of $98{,}765$ again and again until only one digit is left. Which digit do you get?
Answer: 8
$9 + 8 + 7 + 6 + 5 = 35$ and $3 + 5 = 8$ . So $98{,}765$ is not divisible by $3$ or $9$ ; it leaves remainder $8$ when divided by $9$ .
8.
The number $6\square51$ is divisible by $9$ . Which digit goes in the box?
Answer: 6
$6 + 5 + 1 = 12$ . The next multiple of $9$ is $18$ , so the box is $18 - 12 = 6$ . Check: $6651 = 9 \times 739$ .
9.
What remainder does $6014$ leave when divided by $3$ ?
Answer: 2
$6 + 0 + 1 + 4 = 11$ and $11 = 3 \times 3 + 2$ , so the remainder is $2$ . Check: $6014 = 3 \times 2004 + 2$ .
10.
How many different digits (0 to 9) can go in the box so that $3\square5$ is divisible by $3$ ?
Answer: 3
$3 + 5 = 8$ , and with one digit the sum can be at most $17$ . The multiples of $3$ in reach are $9$ , $12$ and $15$ , so the box can be $1$ , $4$ or $7$ ( $315$ , $345$ , $375$ ): $3$ digits.
11.
Use the digit sum to show that $741$ is divisible by $3$ , then find $741 \div 3$ .
Answer: 247
1. What is the digit sum of $741$ ? 12
2. What is $12 \div 3$ ? 4
3. Now work out $741 \div 3$ . 247
12.
Use the digit sum to show that $2358$ is divisible by $9$ , then find $2358 \div 9$ .
Answer: 262
1. What is the digit sum of $2358$ ? 18
2. What is $18 \div 9$ ? 2
3. Now work out $2358 \div 9$ . 262
13.
Use the digit sum to find the remainder when $1234$ is divided by $3$ .
Answer: 1
1. What is the digit sum of $1234$ ? 10
2. What is the remainder of $10 \div 3$ ? 1
3. Check: what is $1233 \div 3$ ? 411
14.
Use repeated digit sums to find the remainder when $879{,}999$ is divided by $9$ .
Answer: 6
1. What is the digit sum of $879{,}999$ ? 51
2. Add the digits of $51$ . 6
3. What is the remainder when dividing by $9$ ? 6
4. Is $879{,}999$ divisible by $3$ ? yes
15.
The number $4\square8$ is divisible by $9$ . Find the missing digit and then work out the division.
Answer: 6
1. Add the known digits $4 + 8$ . 12
2. Which multiple of $9$ can you reach by adding one digit? 18
3. Which digit goes in the box? 6
4. Now work out $468 \div 9$ . 52
16.
Split $372$ as $3 \times 99 + 7 \times 9 + (3 + 7 + 2)$ and use it to find the remainder when $372$ is divided by $9$ .
Answer: 3
1. What is $3 \times 99$ ? 297
2. What is $7 \times 9$ ? 63
3. What is the digit sum $3 + 7 + 2$ ? 12
4. $297$ and $63$ are multiples of $9$ . What remainder does $372$ leave when divided by $9$ ? 3
17.
Check $4782 + 3569 = 8351$ by casting out nines.
Answer: 8
1. Reduce $4782$ to one digit. 3
2. Reduce $3569$ to one digit. 5
3. Add the two results. 8
4. Reduce the answer $8351$ to one digit. 8
18.
Look at $25\square8$ . How many digits make it divisible by $3$ ? Which digit makes it divisible by $9$ ?
Answer: 3
1. Add the known digits $2 + 5 + 8$ . 15
2. How many digits (0 to 9) make the number divisible by $3$ ? 4
3. Which digit makes it divisible by $9$ ? 3
4. Work out $2538 \div 9$ . 282