Divisibility Rules

Show your work for each problem.

1.
Which number is divisible by $2$ ?
1. a) $35$
2. b) $47$
3. c) $82$
4. d) $91$
Answer: $82$
$82$ ends in $2$ , which is even, so it is divisible by $2$ . The others end in odd digits ( $7$ , $5$ , $1$ ).
2.
Which number is divisible by $5$ ?
1. a) $785$
2. b) $123$
3. c) $892$
4. d) $456$
Answer: $785$
$785$ ends in $5$ , so it is divisible by $5$ . The others end in $3$ , $6$ , and $2$ , which are neither $0$ nor $5$ .
3.
What is the sum of the digits of $372$ ?
Answer: 12
$3 + 7 + 2 = 12$ . This digit sum helps determine divisibility by $3$ and $9$ .
4.
Which number is divisible by $3$ ?
1. a) $250$
2. b) $248$
3. c) $244$
4. d) $246$
Answer: $246$
$2 + 4 + 6 = 12$ , and $12$ is divisible by $3$ , so $246$ is divisible by $3$ . The other digit sums are $10$ ( $244$ ), $14$ ( $248$ ) and $7$ ( $250$ ), and none of them is divisible by $3$ .
5.
Which number is divisible by $10$ ?
1. a) $432$
2. b) $245$
3. c) $310$
4. d) $567$
Answer: $310$
$310$ ends in $0$ , so it is divisible by $10$ . The others end in $5$ , $2$ , and $7$ .
6.
Is $738$ divisible by $3$ ? Work it out step by step.
Answer: Yes: 738 ÷ 3 = 246
1. What is 7 + 3? 10
2. Now add the last digit: what is 10 + 8? 18
3. Is 18 divisible by 3? yes
4. So is 738 divisible by 3? yes
7.
Which number is divisible by $4$ ?
1. a) $524$
2. b) $514$
3. c) $554$
4. d) $534$
Answer: $524$
Look at the last two digits: $24 \div 4 = 6$ , so $524$ is divisible by $4$ . $14$ , $34$ and $54$ each leave a remainder of $2$ when divided by $4$ , so $514$ , $534$ and $554$ are not divisible by $4$ , even though they also end in $4$ .
8.
Which number is divisible by both $2$ AND $3$ (therefore divisible by $6$ )?
1. a) $45$
2. b) $85$
3. c) $52$
4. d) $78$
Answer: $78$
$78$ ends in $8$ (even, so divisible by $2$ ) and $7 + 8 = 15$ (divisible by $3$ ). Therefore $78$ is divisible by $6$ . $45$ is divisible by $3$ but not $2$ . $52$ is divisible by $2$ but not $3$ . $85$ is divisible by $5$ only.
9.
Is $1{,}845$ divisible by $9$ ?
Answer: Yes: 1,845 ÷ 9 = 205
1. What is the digit sum 1 + 8 + 4 + 5? 18
2. What is 18 ÷ 9? 2
3. Is 18 divisible by 9? yes
4. Is 1,845 divisible by 9? yes
10.
What are the last two digits of $2{,}736$ ?
Answer: 36
The last two digits of $2{,}736$ are $36$ . Since $36 \div 4 = 9$ , this number is divisible by $4$ .
11.
Is $456$ divisible by $6$ ?
Answer: Yes: 456 ÷ 6 = 76
1. Is 456 divisible by 2? (Does it end in an even digit?) yes
2. What is 4 + 5 + 6? 15
3. Is 15 divisible by 3? yes
4. Since 456 is divisible by both 2 AND 3, is it divisible by 6? yes
12.
Which number is divisible by $9$ but NOT by $6$ ?
1. a) $270$
2. b) $189$
3. c) $162$
4. d) $234$
Answer: $189$
$189$ : digit sum $1 + 8 + 9 = 18$ (divisible by $9$ ). But $189$ ends in $9$ (odd), so it's NOT divisible by $2$ , therefore NOT by $6$ . The others ( $162$ , $234$ , $270$ ) are all even and divisible by both $2$ and $3$ .
13.
What is the remainder when $3{,}574$ is divided by $4$ ? Use only the last two digits.
Answer: 2
$3{,}574 = 3{,}500 + 74$ . $3{,}500$ is a multiple of $100 = 4 \times 25$ , so it is divisible by $4$ . That leaves $74 \div 4 = 18$ R $2$ . The remainder is $2$ , so $3{,}574$ is not divisible by $4$ .
14.
A school has $342$ students. Can they form teams of exactly $9$ students with no one left over? If so, how many teams?
Answer: 38
1. What is the digit sum of 342? (3 + 4 + 2) 9
2. Is 9 divisible by 9? yes
3. Is 342 divisible by 9? yes
4. How many teams of 9 can they form? 38
15.
Which divisibility rules does $2{,}340$ satisfy: $2$ , $3$ , $4$ , $5$ , $6$ , $9$ , $10$ ?
Answer: 2, 3, 4, 5, 6, 9, 10
1. Does 2,340 end in an even digit? (Is it divisible by 2?) yes
2. What is 2 + 3 + 4 + 0? 9
3. Is the digit sum 9 divisible by 3? yes
4. Are the last two digits (40) divisible by 4? yes
5. Does 2,340 end in 0 or 5? (Divisible by 5?) yes
6. Is it divisible by both 2 AND 3? (Divisible by 6?) yes
7. Is the digit sum 9 divisible by 9? yes
8. Does it end in 0? (Divisible by 10?) yes