Patterns in Multiples
When you list the multiples of any number, interesting patterns appear. The ones digits repeat, the digit sums follow a cycle, and certain multiples are shared by different numbers.
In Class 4, you will explore these patterns for multiples of 2 through 10. Recognising these patterns makes multiplication faster and helps you identify multiples without calculating.
What is Patterns in Multiples - Class 4 Maths (Factors and Multiples)?
A multiple of a number is the product you get when you multiply that number by 1, 2, 3, 4, and so on.
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...
Multiples of 5: 5, 10, 15, 20, 25, 30, ...
Patterns in multiples are the regularities you notice in these lists — in the ones digit, digit sums, or on a number grid.
Types and Properties
Key Patterns:
| Multiples of | Pattern in Ones Digit |
|---|---|
| 2 | Always ends in 0, 2, 4, 6, or 8 (even numbers) |
| 3 | Digit sum is always a multiple of 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Always ends in 0 or 5 |
| 9 | Digit sum is always a multiple of 9. Ones digits cycle: 9,8,7,6,5,4,3,2,1,0 |
| 10 | Always ends in 0 |
Solved Examples
Example 1: Example 1: Pattern in multiples of 2
Problem: List the first 10 multiples of 2. What pattern do you see in the ones digit?
Solution:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Step 1: Ones digits: 2, 4, 6, 8, 0, 2, 4, 6, 8, 0
Step 2: The ones digits repeat in a cycle of 5: 2, 4, 6, 8, 0.
Answer: Multiples of 2 always end in 0, 2, 4, 6, or 8. The pattern repeats every 5 multiples.
Example 2: Example 2: Pattern in multiples of 5
Problem: List multiples of 5 from 5 to 50. What do you notice?
Solution:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Ones digits: 5, 0, 5, 0, 5, 0, 5, 0, 5, 0
Answer: Every multiple of 5 ends in either 0 or 5, alternating.
Example 3: Example 3: Digit sum pattern in multiples of 9
Problem: Find the digit sum of the first 8 multiples of 9. What pattern do you see?
Solution:
9→9, 18→1+8=9, 27→2+7=9, 36→3+6=9, 45→4+5=9, 54→5+4=9, 63→6+3=9, 72→7+2=9
Answer: The digit sum of every multiple of 9 is always 9 (or a multiple of 9 for larger numbers).
Example 4: Example 4: Digit sum pattern in multiples of 3
Problem: Check whether the digit sum of multiples of 3 is always divisible by 3.
Solution:
3→3, 6→6, 9→9, 12→1+2=3, 15→1+5=6, 18→1+8=9, 21→2+1=3, 24→2+4=6
Digit sums: 3, 6, 9, 3, 6, 9, 3, 6 — all divisible by 3 ✓
Answer: Yes, the digit sum of every multiple of 3 is always divisible by 3. The digit sums cycle: 3, 6, 9, 3, 6, 9, ...
Example 5: Example 5: Ones digit pattern in multiples of 4
Problem: List the ones digits of the first 10 multiples of 4.
Solution:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Ones digits: 4, 8, 2, 6, 0, 4, 8, 2, 6, 0
Answer: The ones digits of multiples of 4 cycle as: 4, 8, 2, 6, 0 (repeats every 5).
Example 6: Example 6: Common multiples
Problem: List the first 5 multiples of 3 and the first 5 multiples of 4. Find the common multiples.
Solution:
Step 1: Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Step 2: Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Step 3: Numbers in both lists: 12, 24
Answer: Common multiples of 3 and 4 (up to 30): 12 and 24. The smallest common multiple (LCM) is 12.
Example 7: Example 7: 100 chart pattern
Problem: On a 1–100 number chart, Priya colours all multiples of 6. What pattern does she see?
Solution:
Step 1: Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96
Step 2: On the chart, these form a diagonal pattern stepping down 1 row and right 6 columns.
Step 3: Every coloured number is also a multiple of both 2 and 3.
Answer: Multiples of 6 form a diagonal pattern on the chart. They are all even numbers whose digit sum is divisible by 3.
Example 8: Example 8: Using patterns to check multiples
Problem: Is 138 a multiple of 3? Use the digit sum pattern to check.
Solution:
Step 1: Digit sum of 138 = 1 + 3 + 8 = 12
Step 2: Digit sum of 12 = 1 + 2 = 3
Step 3: 3 is divisible by 3 ✓
Answer: Yes, 138 is a multiple of 3.
Example 9: Example 9: Ones digit pattern in multiples of 7
Problem: Do multiples of 7 have a repeating ones digit pattern?
Solution:
7, 14, 21, 28, 35, 42, 49, 56, 63, 70
Ones digits: 7, 4, 1, 8, 5, 2, 9, 6, 3, 0
Step 1: All 10 digits (0–9) appear exactly once before repeating!
Answer: Yes, multiples of 7 have a ones digit cycle of length 10: 7, 4, 1, 8, 5, 2, 9, 6, 3, 0.
Example 10: Example 10: Multiples of 9 — ones digit decreasing
Problem: List multiples of 9 from 9 to 90. Describe the ones digit pattern.
Solution:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90
Ones digits: 9, 8, 7, 6, 5, 4, 3, 2, 1, 0
Answer: The ones digit decreases by 1 each time, going from 9 down to 0.
Key Points to Remember
- Multiples of 2 always end in 0, 2, 4, 6, or 8 (even numbers).
- Multiples of 5 always end in 0 or 5.
- Multiples of 10 always end in 0.
- The digit sum of multiples of 3 is always divisible by 3.
- The digit sum of multiples of 9 is always 9 (for single-cycle multiples).
- Ones digits of multiples of 9 decrease: 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.
- Common multiples appear in the lists of two or more numbers. The smallest is the LCM.
- Patterns help you quickly check if a number is a multiple without dividing.
Practice Problems
- List the ones digits of the first 10 multiples of 6. Does the pattern repeat?
- Is 234 a multiple of 9? Use the digit sum test.
- What are the first 3 common multiples of 4 and 6?
- List multiples of 8 from 8 to 80. What is the ones digit pattern?
- Is 455 a multiple of 5? How do you know without dividing?
- Find the digit sum of each: 27, 54, 81, 108. What do you notice?
- On a 100 chart, which numbers are coloured if you shade both multiples of 2 and multiples of 5?
Frequently Asked Questions
Q1. What is a multiple of a number?
A multiple of a number is the result of multiplying it by any whole number (1, 2, 3, ...). For example, multiples of 4 are 4, 8, 12, 16, 20, and so on.
Q2. What patterns do multiples of 5 show?
Every multiple of 5 ends in either 0 or 5. The ones digits alternate: 5, 0, 5, 0, 5, 0, ...
Q3. How does the digit sum help identify multiples of 3?
Add all the digits of a number. If the sum is divisible by 3, the original number is a multiple of 3. For example, 42: 4+2=6, and 6 is divisible by 3, so 42 is a multiple of 3.
Q4. What is special about the ones digits of multiples of 9?
The ones digits decrease by 1 each time: 9, 8, 7, 6, 5, 4, 3, 2, 1, 0. Also, the digit sum of every multiple of 9 is always 9.
Q5. What is a common multiple?
A common multiple is a number that appears in the multiple lists of two or more numbers. For example, 12 is a common multiple of 3 and 4.
Q6. How do you find the LCM using patterns?
List multiples of both numbers until you find the first number that appears in both lists. That number is the LCM. For 3 and 5: multiples of 3 are 3,6,9,12,15; multiples of 5 are 5,10,15. LCM = 15.
Q7. Do all numbers have ones-digit patterns in their multiples?
Yes. The ones digits of multiples of any number eventually repeat in a cycle. The cycle length varies: multiples of 2 and 5 have a cycle of 5; multiples of 7 have a cycle of 10.
Q8. How are patterns in multiples useful?
They help you quickly test divisibility, predict future multiples, find common multiples, and solve problems without long calculations.
Related Topics
- Multiples of a Number
- Number Patterns (Grade 4)
- Factors of a Number
- Common Factors
- Common Multiples
- HCF (Highest Common Factor) Introduction
- LCM (Least Common Multiple) Introduction
- Prime and Composite Numbers
- Sieve of Eratosthenes
- Even and Odd Number Properties
- Factors and Multiples Word Problems
- Prime Numbers up to 100
