Tests of divisibility
🎯 In this topic you will
- Learn and use tests of divisibility for $4$ and $8$.
🧠 Key Words
- divisible
- divisibility test
- Venn diagram
Show Definitions
- divisible: A number is divisible by another number if it can be divided exactly with no remainder.
- divisibility test: A quick rule used to determine whether one number divides into another without carrying out full division.
- Venn diagram: A visual diagram that uses overlapping circles to show relationships between different sets of numbers or objects.
Using Divisibility Tests
Y ou can use divisibility tests to find out if one number can be divided by another number without having to do the division calculation.
Prime Numbers Explained
A number that is only divisible by 1 and the number itself is called a prime number. You learnt about prime numbers in Unit 3.
💡 Quick Math Tip
Divisibility checks save time: You can use divisibility tests to quickly decide whether a number can be divided exactly by another number without doing the full division.
❓ EXERCISES
1. Look at this set of numbers.
a. Write the numbers that are divisible by $2$.
b. Write the numbers that are divisible by $4$.
👀 Show answer
a. $366$, $422$, $14432$, $790124$, $234444$, $160$, $146$
b. $14432$, $790124$, $234444$, $160$
2. Write down the numbers from this list that are divisible by $4$.
$113,\; 342,\; 632,\; 218,\; 488,\; 784$
How do you know they are divisible by $4$?
Check your answer with your partner.
👀 Show answer
3. Copy and complete the Venn diagram to show where these numbers go.
$304,\; 25,\; 203,\; 400,\; 205,\; 52,\; 502$
👀 Show answer
Divisible by $4$: $304,\; 400,\; 52$
Divisible by $5$: $25,\; 400,\; 205$
Intersection: $400$
4. This sequence shows multiples of $4$.
$4,\; 8,\; 12,\; 16,\; 20,\; \dots$
Will $114$ be in the sequence?
Explain how you know.
👀 Show answer
No. $114 \div 4 = 28.5$, so it is not a multiple of $4$.
5. Here is a number grid.
a. List all the multiples of $4$ that are on the grid.
b. List all the multiples of $8$ that are on the grid.
👀 Show answer
a. $152,\; 156,\; 160,\; 164,\; 168,\; 172,\; 176,\; 180$
b. $152,\; 160,\; 168,\; 176$
6. Copy and complete the Venn diagram to show where these numbers go.
$24302,\; 56824,\; 987204,\; 43200,\; 12404,\; 969696$
👀 Show answer
Divisible by $4$: $56824,\; 987204,\; 43200,\; 12404,\; 969696$
Divisible by $8$: $56824,\; 43200,\; 969696$
7.
a. Write down a number which is divisible by $4$ and $8$.
b. Write down a number which is divisible by $4$ and $5$.
c. Write down a number that is divisible by $2$, $4$, $5$, $10$ and $100$.
👀 Show answer
a. $32$
b. $20$
c. $100$
🧠 Think like a Mathematician
Investigation Question: Is it always, sometimes, or never true that the sum of four even numbers is divisible by $8$?
You will work through this question independently and show convincing reasoning when explaining your findings.
Method:
- Choose any four even numbers and add them together. Record the total.
- Repeat the process with several different sets of four even numbers, including small, large, and mixed-size even values.
- For each total, check whether it is divisible by $8$ by performing the division or using a divisibility rule.
- Look for a pattern: does divisibility by $8$ depend on which even numbers you choose?
- Use your results to decide whether the statement is always, sometimes, or never true.
- Write a convincing explanation for your conclusion using mathematical reasoning.
Follow-up Questions:
Show Answers
- 1: The sum of four even numbers is sometimes divisible by $8$. Some totals (e.g., $2+4+6+8=20$) are not divisible by $8$, while others (e.g., $8+10+12+14=44$) are divisible by $4$ but not $8$. Divisibility by $8$ depends on the specific numbers chosen.
- 2: Any even number can be written as $2k$. The sum of four even numbers is therefore:
$2a + 2b + 2c + 2d = 2(a+b+c+d)$This shows the total is always divisible by $2$ and always even, but not necessarily divisible by $8$.
- 3: Using algebra:
$2(a+b+c+d)$ is divisible by $8$ exactly when $a+b+c+d$ is divisible by $4$. Therefore, the sum of four even numbers is divisible by $8$ only when the sum of the corresponding whole numbers $a+b+c+d$ is a multiple of $4$.This confirms the statement is **sometimes** true.