Divisibility by 4, 6, and 8

Show your work for each problem.

1.
Which number is divisible by $4$ ?
1. a)918
2. b)716
3. c)334
4. d)526
Answer: 716
$716$ ends in $16 = 4 \times 4$ , so it is divisible by $4$ . $34$ , $26$ and $18$ each leave remainder $2$ .
2.
$1000 = 8 \times \square$ . What number goes in the box?
Answer: 125
$1000 \div 8 = 125$ , so $1000 = 8 \times 125$ . That is why only the last three digits matter for $8$ .
3.
Show that $6148$ is divisible by $4$ and find $6148 \div 4$ .
Answer: 1537
1. Which number do the last two digits of 6148 form? 48
2. What is 48 ÷ 4? 12
3. 48 is divisible by 4, so 6148 is too. What is 6148 ÷ 4? 1537
4.
Which rule tells you whether a number is divisible by $8$ ?
1. a)The last two digits form a number divisible by 8
2. b)The last three digits form a number divisible by 8
3. c)The digit sum is divisible by 8
4. d)The last digit is even
Answer: The last three digits form a number divisible by 8
$1000 = 8 \times 125$ but $100 = 8 \times 12 + 4$ , so you need the last three digits. Example: $3124$ ends in $24$ , but $124$ leaves remainder $4$ .
5.
What is the remainder when $9234$ is divided by $4$ ?
Answer: 2
$9200$ is a multiple of $4$ , so only $34$ counts: $34 = 4 \times 8 + 2$ . The remainder is $2$ .
6.
Is $2736$ divisible by $8$ ? Use the last three digits, then find $2736 \div 8$ .
Answer: 342
1. Which number do the last three digits form? 736
2. What is 736 ÷ 8? 92
3. So 2736 is divisible by 8. What is 2736 ÷ 8? 342
7.
Which number is divisible by $6$ ?
1. a)1035
2. b)2346
3. c)4072
4. d)5114
Answer: 2346
$2346$ is even and $2 + 3 + 4 + 6 = 15$ is divisible by $3$ , so $2346 \div 6 = 391$ . $4072$ (digit sum $13$ ) and $5114$ (digit sum $11$ ) fail the test for $3$ ; $1035$ is odd.
8.
What is the smallest digit that can go in the box so that $73\square$ is divisible by $4$ ?
Answer: 2
With an odd tens digit the last digit must be $2$ or $6$ . The smallest is $2$ : $32 = 4 \times 8$ , so $732 \div 4 = 183$ .
9.
Is $4326$ divisible by $6$ ? Test $2$ and $3$ , then find $4326 \div 6$ .
Answer: 721
1. What is the last digit of 4326? (It must be even.) 6
2. What is the digit sum of 4326? 15
3. 6 is even and 15 is divisible by 3. What is 4326 ÷ 6? 721
10.
Use the tens digit shortcut to decide whether $9374$ is divisible by $4$ , then find the remainder.
Answer: 2
1. What is the tens digit of 9374? 7
2. The tens digit is odd, so the last digit must be 2 or 6. What is the last digit of 9374? 4
3. So 9374 is not divisible by 4. What is the remainder of 74 ÷ 4? 2
11.
A whole number is divisible by $8$ . Which statement must be true?
1. a)It is divisible by 6
2. b)It is divisible by 4
3. c)It is odd
4. d)It is divisible by 16
Answer: It is divisible by 4
$8 = 4 \times 2$ , so every multiple of $8$ is a multiple of $4$ . The number $8$ itself shows that the other statements can fail: $8$ is not divisible by $6$ or $16$ , and it is even.
12.
What is the largest digit that can go in the box so that $51\square$ is divisible by $6$ ?
Answer: 6
For $3$ : $6 + \square$ divisible by $3$ gives $0$ , $3$ , $6$ or $9$ . For $2$ the digit must be even: $0$ or $6$ . The largest is $6$ : $516 \div 6 = 86$ .
13.
A shop packs $3480$ pencils in boxes of $8$ . Check with the rule for $8$ that none are left over, then find the number of boxes.
Answer: 435
1. Which number do the last three digits of 3480 form? 480
2. What is 480 ÷ 8? 60
3. So no pencils are left over. How many boxes are there? 435
14.
Which of these years is a leap year?
1. a)2000
2. b)1900
3. c)2100
4. d)1800
Answer: 2000
All four are century years, so they need $400$ . Only $2000 = 400 \times 5$ passes. $1900$ , $2100$ and $1800$ are divisible by $4$ but not by $400$ .
15.
What is the remainder when $74{,}350$ is divided by $8$ ?
Answer: 6
$74{,}000$ is a multiple of $8$ (because $1000 = 8 \times 125$ ), so only $350$ counts: $350 = 8 \times 43 + 6$ . The remainder is $6$ .
16.
Decide whether $2100$ will be a leap year and how many days February $2100$ will have.
Answer: 28
1. What is 2100 ÷ 4? 525
2. 2100 ends in 00, so test 400. What is the remainder of 2100 ÷ 400? 100
3. The remainder is not 0, so 2100 is not a leap year. How many days will February 2100 have? 28
17.
Find the largest digit that can go in the box so that $74\square6$ is divisible by $8$ .
Answer: 9
1. How many digits at the end decide divisibility by 8? 3
2. Try the digit 9: what is 496 ÷ 8? 62
3. 496 is divisible by 8. What is the largest digit that fits in the box? 9
18.
A farm has $5760$ eggs. Show that they fit exactly in boxes of $6$ and also in boxes of $8$ , then find how many boxes of $8$ it can fill.
Answer: 720
1. 5760 is even. What is its digit sum? 18
2. 18 is divisible by 3, so 5760 is divisible by 6. What is 5760 ÷ 6? 960
3. Now the rule for 8: what is 760 ÷ 8? 95
4. 760 is divisible by 8, so 5760 is too. How many boxes of 8? 720