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Teacher Guide: Divisibility by 4, 6, and 8

Test any whole number for 4, 6 and 8 in seconds: read the last two or three digits, or combine the rules for 2 and 3, and see why it works.

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10 questions on Divisibility. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Test whether a whole number is divisible by 4 using its last two digits, including the tens digit shortcut
  • Test whether a whole number is divisible by 8 using its last three digits
  • Test divisibility by 6 by combining the rules for 2 and 3, and explain why both are needed
  • Explain the rules for 4 and 8 with 100 = 4 × 25 and 1000 = 8 × 125
  • Apply the rules to leap years and to packing problems with groups of 6 and 8
Prerequisites
  • • Divisibility Rules: what divisible and remainder mean
  • • Divisibility by 2, 5, and 10: the last digit rules
  • • Divisibility by 3 and 9: the digit sum test for 3
  • • Place value: ones, tens, hundreds, thousands
Discussion Starters
  • 1. Why is the rule for 4 about two digits, the rule for 8 about three digits, but the rule for 2 about only one digit?
  • 2. What would a rule for 16 look like? How many digits would you need to check?
  • 3. Every multiple of 8 is a multiple of 4. Why is the reverse not true?
  • 4. Why can we test 6 with the rules for 2 and 3, but not 8 with the rules for 2 and 4?
  • 5. Someone was born on 29 February 2000. How often do they get a real birthday?
Common Misconceptions

A number ending in 4 or 8 is always divisible by 4 (like 34 or 38)

Remediation: List the multiples of 4 up to 100 and look at their endings: 04, 08, 12, 16, 20... Students see that the last digit alone does not decide, then discover the tens digit pattern (even tens: 0, 4, 8; odd tens: 2, 6).

Divisible by 2 and by 4 means divisible by 8, just like 2 and 3 give 6

Remediation: Test 12, 20 and 28. Then compare factor pairs: 2 and 3 have no common factor, 2 and 4 share 2. Introduce the word coprime.

Every year divisible by 4 is a leap year

Remediation: Show 1900 on an old calendar: February had 28 days. Explain the 400 rule with the average year length 365.2425 days.

Differentiation Ideas

For Struggling Students:

  • • Start with the multiples of 4 up to 100 on a hundred square and colour them to see the pattern in the last two digits
  • • Use only four-digit numbers and one rule at a time
  • • Let students check every verdict with a calculator to build trust in the rules

For On-Level Students:

  • • Mix tests for 4, 6 and 8 in one set and ask students to name the rule they used
  • • Solve missing digit problems, like the smallest digit that makes 73_ divisible by 4
  • • Count the leap years in a century

For Advanced Students:

  • • Find and prove a rule for 16 (last four digits, because 10,000 = 16 × 625)
  • • Explain the halving trick for 8 and why three halvings are enough
  • • Investigate which other divisors can be tested by combining two coprime rules
Standards Alignment
  • 4.OA.B.4 (CCSS.MATH.CONTENT.4.OA.B.4)

    Find all factor pairs for a whole number. Recognize that a whole number is a multiple of each of its factors.

  • 4.NBT.B.6 (CCSS.MATH.CONTENT.4.NBT.B.6)

    Find whole-number quotients and remainders using strategies based on place value.

  • 6.NS.B.4 (CCSS.MATH.CONTENT.6.NS.B.4)

    Find the greatest common factor and least common multiple of two whole numbers and use the distributive property with common factors.

Lesson Resources
  • visualDivisibility explainer

    Shows 45,136 = 45 × 1000 + 136 and highlights the digits that each rule for 2, 4 and 8 reads

  • gameGuess 2, 3 and 6

    Students guess yes or no for 2, 3 and 6 on each number, then reveal the rule and score

  • activityLeap year hunt

    Students list the birth years of family members and test which ones were leap years

  • worksheetPacking problems

    Orders of eggs, muffins and bottles to pack in boxes of 6 and crates of 8

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A divisibility rule tells you whether a number is divisible by another number (division with remainder 00) without doing the whole division. In this lesson you learn the rules for 44, 88 and 66.

Divisible by 4: the last two digits

A whole number is divisible by 44 when the number formed by its last two digits is divisible by 44.
7316:16=4×4  ⇒  7316÷4=18297316: \quad 16 = 4 \times 4 \;\Rightarrow\; 7316 \div 4 = 1829
Why? 7316=73×100+167316 = 73 \times 100 + 16, and 100=4×25100 = 4 \times 25. Every whole number of hundreds is a multiple of 44, so only the last two digits can leave a remainder.
Quick check with the tens digit: if the tens digit is even, the last digit must be 00, 44 or 88. If the tens digit is odd, the last digit must be 22 or 66.

Divisible by 8: the last three digits

A whole number is divisible by 88 when the number formed by its last three digits is divisible by 88, because 1000=8×1251000 = 8 \times 125.
45,136:136=8×17  ⇒  45,136÷8=564245{,}136: \quad 136 = 8 \times 17 \;\Rightarrow\; 45{,}136 \div 8 = 5642

Divisible by 6: the rules for 2 and 3 together

A whole number is divisible by 66 when it is divisible by 22 and by 33: the last digit is even and the digit sum is divisible by 33. This works because 6=2×36 = 2 \times 3 and 22 and 33 are coprime.
5742:last digit 2 (even),5+7+4+2=18=3×6  ⇒  5742÷6=9575742: \quad \text{last digit } 2 \text{ (even)}, \quad 5 + 7 + 4 + 2 = 18 = 3 \times 6 \;\Rightarrow\; 5742 \div 6 = 957

A chain of rules

divisible by 8  ⇒  divisible by 4  ⇒  divisible by 2\text{divisible by } 8 \;\Rightarrow\; \text{divisible by } 4 \;\Rightarrow\; \text{divisible by } 2
The chain only goes one way: 1212 is divisible by 44 but not by 88.

Worked Examples

Is 73167316 divisible by 44?
1

Split off the last two digits

7316=73×100+167316 = 73 \times 100 + 16 → Last two digits: 1616
2

Remember why the hundreds do not matter

100=4×25100 = 4 \times 25, so 73007300 is a multiple of 44 → Only 1616 decides
3

Test the last two digits

16=4×416 = 4 \times 4 → 1616 is divisible by 44
4

Conclude and check

7316÷4=18297316 \div 4 = 1829 → Divisible, remainder 00

Common Mistakes

Checking only the last digit for 44: "21342134 ends in 44, so it is divisible by 44"

Why it's wrong: The rule for 44 reads the last two digits, because 1010 is not a multiple of 44. Here 34=4×8+234 = 4 \times 8 + 2.

Correct: Test the last two digits: 3434 leaves remainder 22, so 21342134 is not divisible by 44.

Checking only the last two digits for 88: "31243124 ends in 24=8×324 = 8 \times 3, so it is divisible by 88"

Why it's wrong: 100100 is not a multiple of 88 (100=8×12+4100 = 8 \times 12 + 4), so the hundreds digit still matters. The rule for 88 needs the last three digits.

Correct: Test 124124: 124=8×15+4124 = 8 \times 15 + 4. So 31243124 is not divisible by 88 (remainder 44).

Thinking that divisible by 22 and by 44 means divisible by 88

Why it's wrong: 1212 is divisible by 22 and by 44, but 12÷8=1.512 \div 8 = 1.5. Combining rules only works when the two numbers are coprime, like 22 and 33 for 66. 22 and 44 share the factor 22.

Correct: For 88 use its own rule (last three digits). For 66 you may combine 22 and 33.

Testing only one part of the rule for 66: "2727 is divisible by 33, so it is divisible by 66"

Why it's wrong: 66 needs both tests. 2727 is odd, so it fails the test for 22. Likewise 2626 is even but its digit sum 88 fails the test for 33.

Correct: A number is divisible by 66 only if it is even and its digit sum is divisible by 33, for example 57425742.

Why It Matters

These three rules show up whenever things come in groups of 4, 6 or 8:
  • Calendars: a year is a leap year when it is divisible by 44, with a special rule for century years such as 19001900 and 20002000
  • Packing: eggs and drinks are sold in packs of 66, crayons and bottles in boxes or crates of 88. Can 17341734 muffins fill boxes of 66 exactly?
  • Computers: 11 byte is 88 bits, and memory sizes are multiples of 88
  • Fractions: spotting a factor 44, 66 or 88 in a numerator and denominator lets you simplify quickly, like 2432=34\frac{24}{32} = \frac{3}{4} after dividing both by 88
  • Checking work: a quick look at the last digits tells you if an answer like 7316÷47316 \div 4 can come out whole before you calculate it
The ideas behind the rules (100=4×25100 = 4 \times 25, 1000=8×1251000 = 8 \times 125, 6=2×36 = 2 \times 3) are also the first step toward the rules for 12, 15, 18 and beyond.

Real World Applications

Leap years in a century

The calendar adds a day (29 February) in every year divisible by 4, but skips century years that are not divisible by 400.

Example:

20242024 is a leap year (24=4×624 = 4 \times 6), 20002000 was one (2000=400×52000 = 400 \times 5), but 19001900 was not (1900÷400=4.751900 \div 400 = 4.75).

1Try It Yourself

You count the leap years from 2001 to 2100, including 2100.

How many leap years are there in this period?

Step 1: Write the mathematical expression

Type the number of leap years:

Packing in sixes

Bakeries, egg farms and drink makers pack in 6s. The rules for 2 and 3 tell them before counting whether nothing will be left over.

Example:

43204320 eggs fill boxes of 66 exactly: 43204320 is even and 4+3+2+0=94 + 3 + 2 + 0 = 9 is divisible by 33, so 4320÷6=7204320 \div 6 = 720 boxes.

2Try It Yourself

A bakery has 1734 muffins and packs them in boxes of 6.

How many full boxes can it fill, with no muffins left over?

Step 1: Write the mathematical expression

Write the division that gives the number of boxes:

Crates of eight

Bottles and cans often travel in crates of 8. Only the last three digits of a large order decide whether the crates come out even.

Example:

15601560 pencils fit in boxes of 88: the last three digits 560=8×70560 = 8 \times 70, so 1560÷8=1951560 \div 8 = 195 boxes.

3Try It Yourself

A warehouse receives 23,512 bottles to load into crates of 8.

How many full crates does it fill?

Step 1: Write the mathematical expression

Write the division that gives the number of crates:

Key Takeaways

  • 1Divisible by 44: the last two digits form a number divisible by 44, because 100=4×25100 = 4 \times 25
  • 2Tens digit shortcut for 44: even tens digit needs last digit 00, 44 or 88; odd tens digit needs 22 or 66
  • 3Divisible by 88: the last three digits form a number divisible by 88, because 1000=8×1251000 = 8 \times 125
  • 4Divisible by 66: even and digit sum divisible by 33 (both tests must pass)
  • 5Divisible by 88 implies divisible by 44, which implies divisible by 22, but not the other way round (1212)
  • 6Leap years: divisible by 44, but century years only when divisible by 400400 (20002000 yes, 19001900 no)

Frequently Asked Questions

Why do the last two digits decide divisibility by 4?

Any whole number is a number of hundreds plus its last two digits: 7316=73×100+167316 = 73 \times 100 + 16. Since 100=4×25100 = 4 \times 25, every whole number of hundreds is divisible by 44. The remainder of the whole number divided by 44 is therefore the same as the remainder of the last two digits. The same idea with 1000=8×1251000 = 8 \times 125 gives the rule for 88.

If a number is divisible by 4, is it also divisible by 8?

Not always. 1212, 2020 and 2828 are divisible by 44 but not by 88. The chain only goes downward: every multiple of 88 is a multiple of 44 (because 8=4×28 = 4 \times 2), and every multiple of 44 is even.

Can I build other rules the way 6 = 2 and 3 works?

Yes, but only with coprime numbers (no common factor other than 11). 6=2×36 = 2 \times 3 works because 22 and 33 are coprime. 8=2×48 = 2 \times 4 does not work, because 22 and 44 share the factor 22: 1212 passes both tests and is still not divisible by 88. The lesson Combined Divisibility Rules shows this for 1212, 1515, 1818 and 3636.

Why must century years be divisible by 400 to be leap years?

A year lasts about 365.2422365.2422 days. Adding a leap day every 44 years gives an average of 365.25365.25 days, a bit too long. Skipping 33 leap days every 400400 years (17001700, 18001800, 19001900, but not 20002000) leaves 9797 leap years in 400400 years: 365+97400=365.2425365 + \frac{97}{400} = 365.2425 days, very close to the real value.

Glossary

Divisibility rule
A shortcut that tells whether a number is divisible by another without doing the full division, for example the last two digits for 44.
Divisible by
A whole number is divisible by nn when dividing it by nn leaves remainder 00: 73167316 is divisible by 44.
Remainder
What is left after dividing as many whole times as possible: 34÷4=834 \div 4 = 8 remainder 22.
Digit sum
The sum of all digits of a number: the digit sum of 57425742 is 5+7+4+2=185 + 7 + 4 + 2 = 18.
Coprime
Two whole numbers are coprime when their only common factor is 11, like 22 and 33. Then divisible by both means divisible by their product.
Leap year
A year with 366366 days (29 February added): divisible by 44, but a century year only when divisible by 400400.

Formula Card

Divisible by 4

Last two digits divisible by 44

Test the last two digits, because $100 = 4 \times 25$

Divisible by 8

Last three digits divisible by 88

Test the last three digits, because $1000 = 8 \times 125$

Divisible by 6

Divisible by 22 and by 33

Last digit even and digit sum divisible by 3

Leap year

Divisible by 44; century years: by 400400

2024 and 2000 are leap years, 1900 and 2100 are not

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