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Teacher Guide: Divisibility by 2, 5, and 10

Learn the quick rules to tell if a number is divisible by 2, 5, or 10 just by looking at its last digit.

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Class quiz

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

For Teachers

Learning Objectives
  • State the divisibility rules for 2, 5, and 10
  • Apply divisibility rules to determine if a number is divisible by 2, 5, or 10
  • Explain why the last digit determines divisibility by 2, 5, and 10
  • Solve real-world problems using divisibility rules
Prerequisites
  • • Understanding of division as equal sharing
  • • Recognizing even and odd numbers
  • • Identifying place values (ones, tens, hundreds)
Discussion Starters
  • 1. Why do you think the last digit is so important for these rules?
  • 2. Can a number be divisible by 5 but not by 10? Give an example.
  • 3. If you know a number is divisible by 10, what else do you know about it?
  • 4. How could divisibility rules help you at a store or when sharing snacks?
Common Misconceptions

All numbers ending in 5 are divisible by 10

Remediation: Show that 15, 25, 35 end in 5 but give decimals when divided by 10. Only 0 works for divisibility by 10.

Large numbers need special rules

Remediation: Test with 1,234,567,890. It ends in 0, so it's divisible by 2, 5, and 10. The rules work regardless of number size.

Differentiation Ideas

For Struggling Students:

  • • Use physical manipulatives to show equal grouping
  • • Focus on two-digit numbers first
  • • Create a reference chart with the three rules

For On-Level Students:

  • • Apply rules to three and four-digit numbers
  • • Solve word problems involving divisibility
  • • Explore why these rules work mathematically

For Advanced Students:

  • • Investigate combined divisibility (e.g., divisible by both 2 and 5)
  • • Discover rules for 4 and 25 (related to these rules)
  • • Create their own divisibility problems
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.

Lesson Resources
  • visualLast Digit Detector

    Highlight the last digit and apply divisibility rules

  • activitySort the Numbers

    Categorize numbers by divisibility by 2, 5, and 10

  • worksheetDivisibility Practice

    Apply rules to lists of numbers

Lesson Content

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

Definition

Divisibility rules are shortcuts that help us quickly determine if one number divides evenly into another without doing the actual division.

Divisibility by 2

A number is divisible by 2 if its last digit is even: 0,2,4,6,0, 2, 4, 6, or 88.
24÷2=12✓24 \div 2 = 12 \quad \checkmark
37÷2=18.5(not a whole number)37 \div 2 = 18.5 \quad \text{(not a whole number)}

Divisibility by 5

A number is divisible by 5 if its last digit is 00 or 55.
45÷5=9✓45 \div 5 = 9 \quad \checkmark
73÷5=14.6(not a whole number)73 \div 5 = 14.6 \quad \text{(not a whole number)}

Divisibility by 10

A number is divisible by 10 if its last digit is 00.
120÷10=12✓120 \div 10 = 12 \quad \checkmark
85÷10=8.5(not a whole number)85 \div 10 = 8.5 \quad \text{(not a whole number)}

Worked Examples

Is 1,458 divisible by 2?
1

Look at the last digit

The last digit of 1,4581{,}458 is 88 → Last digit: 88
2

Check if it's even

Is 88 in the set {0,2,4,6,8}\{0, 2, 4, 6, 8\}? → Yes, 88 is even
3

Apply the rule

Since the last digit is even, 1,4581{,}458 is divisible by 22 → 1,458÷2=7291{,}458 \div 2 = 729

Common Mistakes

Looking at the first digit instead of the last digit

Why it's wrong: For divisibility by 2, 5, and 10, only the last digit matters. The first digit tells us the size of the number, not its divisibility.

Correct: Always look at the rightmost (last) digit: in 847, check 7, not 8.

Thinking a number ending in 5 is divisible by 10

Why it's wrong: Divisibility by 10 is stricter than by 5. A number must end in 0 (not just 0 or 5) to be divisible by 10.

Correct: 75 is divisible by 5 but NOT by 10. Only 70, 80, 90... are divisible by 10.

Confusing 'even' with 'divisible by 10'

Why it's wrong: All numbers divisible by 10 are even, but not all even numbers are divisible by 10.

Correct: 24 is even (divisible by 2) but not divisible by 10. 20 is both even AND divisible by 10.

Why It Matters

These divisibility rules save time and help you work smarter with numbers:
  • Mental math: Quickly check if you can split things evenly without a calculator
  • Simplifying fractions: Know instantly if you can divide both numerator and denominator by 2, 5, or 10
  • Real-world sharing: Can 24 cookies be shared equally between 2 friends? Can 35 students form groups of 5?
  • Building blocks: These simple rules prepare you for more complex divisibility tests (3, 4, 6, 9)
Mastering these three rules makes all your future math work faster!

Real World Applications

Sharing Equally

When splitting items between people, divisibility rules help you know instantly if it works.

Example:

Can 36 candies be shared equally between 2 children? Yes! 3636 ends in 66 (even), so 36÷2=1836 \div 2 = 18 each.

1Try It Yourself

You have 125 stickers to put in packs of 5.

Can you pack them all with none left over, and how many packs do you get?

Step 1: Write the mathematical expression

Write the division that shares 125 stickers into packs of 5:

Counting Money

Money often uses denominations of 10, making divisibility by 10 very practical.

Example:

If you have 240 cents, can you exchange them for dimes (10 cents)? Yes! 240240 ends in 00, so 240÷10=24240 \div 10 = 24 dimes.

2Try It Yourself

A piggy bank has 375 cents.

Can these be exchanged for an exact number of dimes?

Step 1: Write the mathematical expression

Write the division that tells how many dimes 375 cents make:

Key Takeaways

  • 1A number is divisible by 2 if its last digit is even (0,2,4,6,80, 2, 4, 6, 8)
  • 2A number is divisible by 5 if its last digit is 00 or 55
  • 3A number is divisible by 10 if its last digit is 00
  • 4Every number divisible by 10 is also divisible by both 2 and 5
  • 5Only the last digit matters for these three rules

Frequently Asked Questions

Why does the last digit rule work?

Any number can be written as a multiple of 10 plus its last digit. For example, 347=340+7347 = 340 + 7. Since 340340 is a multiple of 10, it is divisible by 2, 5, and 10, so we only need to check if the last digit (77) is divisible.

If a number is divisible by 10, is it always divisible by 2 and 5?

Yes! Since 10=2×510 = 2 \times 5, any number divisible by 10 must also be divisible by both 2 and 5. For example, 60 is divisible by 10, 5, and 2.

What about large numbers like 1,000,000?

The same rules apply! 1,000,000 ends in 0, so it's divisible by 2, 5, AND 10. These rules work for numbers of any size.

Glossary

Divisible
A number is divisible by another if dividing them gives a whole number with no remainder
Even number
A number divisible by 2, ending in 0, 2, 4, 6, or 8
Odd number
A number not divisible by 2, ending in 1, 3, 5, 7, or 9
Last digit
The digit in the ones place (rightmost position)
Remainder
The amount left over after division

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