Combined Divisibility Rules

Show your work for each problem.

1.
Which pair of tests together proves that a number is divisible by $12$ ?
1. a)2 and 6
2. b)3 and 4
3. c)3 and 6
4. d)2 and 3
Answer: 3 and 4
$12 = 3 \times 4$ and 3 and 4 are coprime, so passing both tests proves divisibility by 12. 2 and 6 share the factor 2 (18 passes both but is not divisible by 12), 2 and 3 only prove divisibility by 6, and 3 and 6 also only prove 6.
2.
A number is divisible by $2$ and by $9$ . Which number between $10$ and $20$ must also divide it?
Answer: 18
2 and 9 have no common factor except 1, so the number is divisible by $2 \times 9 = 18$ .
3.
Is $2412$ divisible by $12$ ? Use the tests for 3 and 4, then find the quotient.
Answer: 201
1. What is the digit sum of 2412? 9
2. Which number do the last two digits form? 12
3. Both tests pass. What is $2412 \div 12$ ? 201
4.
Which number is divisible by $11$ ?
1. a)3542
2. b)9091
3. c)4825
4. d)6170
Answer: 3542
For $3542$ : $2 - 4 + 5 - 3 = 0$ , a multiple of 11, so $3542 = 11 \times 322$ . The others give $7$ , $-12$ and $-17$ .
5.
Find the alternating sum of $2816$ , starting with $+$ at the ones digit: $6 - 1 + 8 - 2$ .
Answer: 11
$6 - 1 + 8 - 2 = 11$ . Since 11 is a multiple of 11, $2816$ is divisible by 11: $2816 = 11 \times 256$ .
6.
Is $735$ divisible by $15$ ? Use the tests for 5 and 3, then find the quotient.
Answer: 49
1. What is the last digit of 735? 5
2. What is the digit sum of 735? 15
3. Both tests pass. What is $735 \div 15$ ? 49
7.
$18$ is divisible by $2$ and by $6$ , but not by $12$ . Why?
1. a)2 and 6 share the factor 2
2. b)18 is smaller than 2 × 12
3. c)6 is not a factor of 12
4. d)12 has no divisibility rule
Answer: 2 and 6 share the factor 2
2 and 6 share the factor 2, so they are not coprime. Passing both tests only proves divisibility by $\text{lcm}(2, 6) = 6$ , not by 12.
8.
Find the digit $d$ that makes $7d4$ divisible by $18$ .
Answer: 7
$18 = 2 \times 9$ . The number ends in 4, so it is even. The digit sum $7 + d + 4 = 11 + d$ must be a multiple of 9, so $11 + d = 18$ and $d = 7$ . Check: $774 \div 18 = 43$ .
9.
Is $61{,}809$ divisible by $11$ ? Use the alternating sum, then find the quotient.
Answer: 5619
1. Add the digits in the 1st, 3rd and 5th places from the right. 23
2. Add the digits in the 2nd and 4th places from the right. 1
3. What is the difference of the two sums? 22
4. So 11 divides 61,809. What is $61{,}809 \div 11$ ? 5619
10.
Is $4374$ divisible by $18$ ? Use the tests for 2 and 9, then find the quotient.
Answer: 243
1. What is the last digit of 4374? 4
2. What is the digit sum of 4374? 18
3. Both tests pass. What is $4374 \div 18$ ? 243
11.
Which number is divisible by $36$ ?
1. a)2466
2. b)2556
3. c)4518
4. d)1236
Answer: 2556
$2556$ : $56 \div 4 = 14$ and $2 + 5 + 5 + 6 = 18$ , so it is divisible by 36 ( $2556 = 36 \times 71$ ). $2466$ and $4518$ fail the test for 4 (66 and 18), $1236$ fails the test for 9 (digit sum 12).
12.
How many of the numbers $2, 3, 4, 5, 6, 8, 9, 10, 11, 12$ divide $4620$ ?
Answer: 8
Ends in 0: 2, 5, 10. Digit sum 12: 3 yes, 9 no. $20 \div 4 = 5$ : 4 yes. $620 \div 8 = 77.5$ : 8 no. Alternating sum $0 - 2 + 6 - 4 = 0$ : 11 yes. Pairs: 6 and 12 yes. So 8 numbers: 2, 3, 4, 5, 6, 10, 11, 12.
13.
$4530$ is divisible by $2$ and by $6$ . Is it divisible by $12$ ? Check with the coprime pair 3 and 4.
Answer: 6
1. What is the digit sum of 4530? 12
2. Which number do the last two digits form? 30
3. What is the remainder of $30 \div 4$ ? 2
4. The test for 4 fails. What is the remainder of $4530 \div 12$ ? 6
14.
Which digit $d$ makes $4d52$ divisible by $36$ ?
1. a)9
2. b)3
3. c)7
4. d)4
Answer: 7
$52 \div 4 = 13$ , so the test for 4 passes for every $d$ . The digit sum $4 + d + 5 + 2 = 11 + d$ must be a multiple of 9, so $d = 7$ . Check: $4752 \div 36 = 132$ .
15.
Quick factor search: how many of the numbers $2, 3, 4, 5, 6, 8, 9, 10, 11, 12$ divide $2520$ ?
Answer: 9
1. What is the digit sum of 2520? 9
2. Test 8 with the last three digits. What is $520 \div 8$ ? 65
3. What is the alternating sum of 2520, starting with + at the ones digit? 1
4. Count the numbers in the list that divide 2520. 9
16.
$33 = 3 \times 11$ . Find the digit $d$ that makes $8d14$ divisible by $33$ .
Answer: 5
For 11: $4 - 1 + d - 8 = d - 5$ must be a multiple of 11, and for a digit only $d = 5$ works. For 3: $8 + 5 + 1 + 4 = 18$ , divisible by 3. So $d = 5$ : $8514 \div 33 = 258$ .
17.
Find the digit $d$ that makes $52d6$ divisible by $12$ .
Answer: 5
1. For how many digits $d$ is $d6$ a multiple of 4? 5
2. What is the digit sum of 52d6 without $d$ ? 13
3. Which odd digit $d$ makes $13 + d$ a multiple of 3? 5
4. Check: what is $5256 \div 12$ ? 438
18.
$132 = 11 \times 12$ . Is $7128$ divisible by $132$ ? Check 11 and 12, then find the quotient.
Answer: 54
1. What is the greatest common factor of 11 and 12? 1
2. What is the digit sum of 7128? 18
3. The last two digits form 28. What is $28 \div 4$ ? 7
4. What is the alternating sum of 7128, starting with + at the ones digit? 0
5. All tests pass. What is $7128 \div 132$ ? 54