Tests of divisibility
🎯 In this topic you will
- Use tests of divisibility for 3, 6, and 9 to determine whether a number is divisible by these values.
🧠 Key Words
- divisible
- factor
- multiple
- test of divisibility
- Venn diagram
Show Definitions
- divisible: A number is divisible by another number if it can be divided exactly with no remainder.
- factor: A number that divides another number exactly without leaving a remainder.
- multiple: A number obtained by multiplying a given number by an integer.
- test of divisibility: A rule that helps determine quickly whether one number can be divided exactly by another without performing full division.
- Venn diagram: A diagram using overlapping circles to show relationships between different sets of numbers or objects.
🔢 Divisibility Rules Make Division Faster
Divisibility rules for whole numbers are useful because they help you quickly find out if a number can be divided without leaving a remainder.
❓ EXERCISES
1. Which of these numbers is divisible by $3$? Explain how you know. $935$, $9203$, $43\,719$
👀 Show answer
$935: 9+3+5=17$ (not divisible by $3$)
$9203: 9+2+0+3=14$ (not divisible by $3$)
$43\,719: 4+3+7+1+9=24$ (divisible by $3$).
Therefore $43\,719$ is divisible by $3$.
2. Find the missing digits to copy and complete the calculations.
a. $1\Box \times 3 = 5 7$
b. $\Box\Box \times 3 = 5 1$
c. $\Box\Box \times 3 = 4\Box$
👀 Show answer
b. $17 \times 3 = 51$
c. Possible answer: $14 \times 3 = 42$
3. Jiao is thinking of a number. She says, ‘My number is between $50$ and $100$. It is divisible by $3$ and $4$. The tens digit is double the ones digit.’ What number is Jiao thinking of?
👀 Show answer
Multiples of $12$ between $50$ and $100$ are $60, 72, 84, 96$.
Only $84$ has a tens digit that is double the ones digit ($8 = 2 \times 4$).
The number is $84$.
4. Start at $99$ and list the next four numbers that are divisible by $9$.
👀 Show answer
$99 + 9 = 108$
$108 + 9 = 117$
$117 + 9 = 126$
$126 + 9 = 135$
The next four numbers are $108, 117, 126, 135$.
5. Find a number between $90$ and $100$ that is divisible by $6$.
👀 Show answer
$96$ works because it is even and $9+6=15$, which is divisible by $3$.
The number is $96$.
6. Copy the Venn diagram. Put these numbers on the diagram. $16, 21, 24, 27, 36$
What do you know about the numbers in the yellow region?
👀 Show answer
$21$: divisible by $3$ only
$24$: divisible by $3$ and $6$
$27$: divisible by $3$ and $9$
$36$: divisible by $3$, $6$, and $9$
Numbers in the yellow region are divisible by both $6$ and $9$.
7. Copy the table. Put ticks in the boxes to show whether these numbers are divisible by $3$, $6$ and $9$.
| Number | $3$ | $6$ | $9$ |
|---|---|---|---|
| $987$ | $✓$ | $✗$ | $✗$ |
| $495$ | $✓$ | $✗$ | $✓$ |
| $3594$ | $✓$ | $✓$ | $✗$ |
👀 Show answer
$495$: divisible by $3$ and $9$.
$3594$: divisible by $3$ and $6$.
8. Oscar is thinking of a number. He says, ‘My number is between $200$ and $220$. It is divisible by $6$. The sum of the digits is $3$.’ What number is Oscar thinking of?
👀 Show answer
$210$ is even and divisible by $3$, so it is divisible by $6$.
The number is $210$.
9. Write a digit in each box so that all the numbers are divisible by $3$.
a. $23\Box$
b. $3\Box5$
c. $83\Box49$
👀 Show answer
b. $3\Box5$: digit $1$, $4$, or $7$ works (sum becomes $9$, $12$, or $15$).
c. $83\Box49$: digit $0$, $3$, $6$, or $9$ works (sum becomes a multiple of $3$).
🧠 Think like a Mathematician
Problem:
Paulo has forgotten the 4-digit number that allows him to open his case. He knows that the number:
- is less than 3000
- is divisible by 3
- has a tens digit that is divisible by 3
- has a hundreds digit that is divisible by 3
- has a ones digit that is bigger than the tens digit
- has a ones digit and a thousands digit that are not multiples of 3
- has no zeros
Task: Find all the numbers that satisfy all these conditions.
👀 show answer
Step-by-step reasoning:
- The thousands digit must be 1 or 2 (because the number is less than 3000 and not a multiple of 3).
- The hundreds and tens digits must be 3, 6, or 9 (multiples of 3).
- The ones digit must be greater than the tens digit and not a multiple of 3.
- The whole number must also be divisible by 3, so the sum of digits must be divisible by 3.
All numbers satisfying the conditions:
1335, 1338, 1635, 1638, 1935, 1938
2334, 2337, 2634, 2637, 2934, 2937
1368, 1668, 1968
2367, 2667, 2967
Total solutions: 18 numbers.