Tests for divisibilty
🎯 In this topic you will
- Use divisibility tests to find factors of large numbers
🧠 Key Words
- divisible
- tests for divisibility
Show Definitions
- divisible: A number is divisible by another if it can be divided exactly with no remainder.
- tests for divisibility: Simple rules used to check whether one number divides evenly into another.
📗 Understanding divisibility and remainders
2, 3 and 5 are all factors of 30.
You say that ‘30 is divisible by 2’ because 30 ÷ 2 does not have a remainder.
30 is divisible by 3 and 30 is divisible by 5.
30 is not divisible by 4 because 30 ÷ 4 = 7 with remainder 2 (which can be written as 7 r 2).
🔎 Reasoning Tip
Divisibility by 2: A whole number is divisible by 2 when 2 is a factor of that number.
🔍 Divisibility rules for numbers
87 654 is a large number.
Is 87 654 divisible by 2? By 3? By 4? By 5?
Here are some rules for divisibility:
- A number is divisible by 2 when the last digit is 0, 2, 4, 6 or 8.
87 654 is divisible by 2 because the last digit is 4. - A number is divisible by 3 when the sum of the digits is a multiple of 3.
8 + 7 + 6 + 5 + 4 = 30 and 30 = 10 × 3, so 87 654 is divisible by 3. - A number is divisible by 4 when the number formed by the last two digits is divisible by 4.
The last two digits of 87 654 are 54 and 54 ÷ 4 = 13 r 2. So 87 654 is not divisible by 4. - A number is divisible by 5 when the last digit is 0 or 5.
The last digit of 87 654 is 4, so it is not divisible by 5. - A number is divisible by 6 when it is divisible by 2 and 3.
87 654 is divisible by 6. - To test for divisibility by 7, remove the last digit, 4, to leave 8765
- Subtract twice the last digit from 8765, that is:
8765 – 2 × 4 = 8765 – 8 = 8757 - If this number is divisible by 7, so is the original number.
8757 ÷ 7 = 1252 with no remainder and so 87 654 is divisible by 7.
- Subtract twice the last digit from 8765, that is:
- A number is divisible by 8 when the number formed by the last three digits is divisible by 8.
654 ÷ 8 = 81 r 6, so 87 654 is not divisible by 8. - A number is divisible by 9 when the sum of the digits is divisible by 9.
8 + 7 + 6 + 5 + 4 = 30 and 30 is not divisible by 9. So 87 654 is not divisible by 9. - A number is divisible by 10 when the last digit is 0.
The last digit of 87 654 is 4, so 87 654 is not divisible by 10. - A number is divisible by 11 when the difference between the sum of the odd digits and the sum of the even digits is 0 or a multiple of 11.
The sum of the odd digits of 87 654 is 4 + 6 + 8 = 18.
The sum of the even digits of 87 654 is 5 + 7 = 12.
18 – 12 = 6, so 87 654 is not a multiple of 11.
❓ EXERCISES
1a. Show that the number 28 572 is divisible by 3 but not by 9.
👀 Show answer
24 is divisible by 3 → so 28 572 is divisible by 3.
24 is not divisible by 9 → so 28 572 is not divisible by 9.
1b. Change the final digit of 28 572 to make a number that is divisible by 9.
👀 Show answer
We need the sum to be 27 (next multiple after 24).
So increase the digit sum by 3 → change final digit from 2 to 5.
New number: 28 575
2a. Show that 57 423 is divisible by 3 but not by 6.
👀 Show answer
Last digit is 3 → not even → not divisible by 2, so not divisible by 6.
2b. The number 5742* is divisible by 6. Find the possible values of the digit *.
👀 Show answer
Try each digit: Sum of 5 + 7 + 4 + 2 + * must be divisible by 3
Sum without * = 18 → valid if * = 0, 3, 6, or 9 → only * = 0 and 6 are even
So possible values: 0 or 6
3a. Show that 25 764 is divisible by 2 and by 4.
👀 Show answer
Last two digits = 64 → divisible by 4 → divisible by both 2 and 4.
3b. Is 25 764 divisible by 8? Give a reason for your answer.
👀 Show answer
764 ÷ 8 = 95.5 → not whole
So 25 764 is not divisible by 8.
4a. Show that 3 and 4 are factors of 25 320.
👀 Show answer
Last two digits = 20 → divisible by 4 →
So 3 and 4 are both factors of 25 320
4b. Find two more factors of 25 320 that are between 1 and 12.
👀 Show answer
Try 8: 320 is divisible by 8 → whole result → ✔️
Two more factors: 5 and 8
5a. Choose any four digits.
👀 Show answer
5b. If it is possible, arrange your digits to make a number that is divisible by:
i) 2 ii) 3 iii) 4 iv) 5 v) 6
👀 Show answer
i) 1236 → even → divisible by 2 ✔️
ii) digit sum = 12 → divisible by 3 ✔️
iii) ends in 36 → divisible by 4 ✔️
iv) ends in 6 → not divisible by 5 ❌
v) divisible by both 2 and 3 → ✔️
5c. Can you arrange your digits to make a number that is divisible by all five numbers in part a? If not, can you make a number that is divisible by four of the numbers?
👀 Show answer
But we can make one divisible by 2, 3, 4, and 6.
Example: 1236 is divisible by 2, 3, 4, and 6.
5d. Give your answers to a partner to check.
👀 Show answer
6a. Show that 924 is divisible by 11.
👀 Show answer
So 924 is divisible by 11.
6b. Is 161 084 divisible by 11? Give a reason for your answer.
👀 Show answer
0 is divisible by 11 → so 161 084 is divisible by 11 ✔️
7a. Use a test for divisibility to show that 2583 is divisible by 7.
👀 Show answer
So 2583 is divisible by 7.
7b. Use a test for divisibility to show that 3852 is not divisible by 7.
👀 Show answer
So 3852 is not divisible by 7.
8a. Show that only two numbers between 1 and 10 are factors of 22 599.
👀 Show answer
22 599 is divisible by 1 and 3 only.
Sum of digits = 2+2+5+9+9 = 27 → divisible by 3 ✔️
So only factors between 1 and 10 are: 1 and 3
8b. What numbers between 1 and 10 are factors of 99 522?
👀 Show answer
Ends in 2 → divisible by 2 ✔️
Not divisible by 4 (last two digits = 22) ❌
Not divisible by 5 (doesn’t end in 0 or 5) ❌
99 522 ÷ 6 = 16 587 → whole ✔️
Divisible by: 1, 2, 3, 6, 9
9. Copy and complete this table. The first line has been done for you.
| Number | Factors between 1 and 10 |
|---|---|
| 12 | 2, 3, 4, 6 |
| 123 | |
| 1234 | |
| 12345 | |
| 123456 |
👀 Show answer
| Number | Factors between 1 and 10 |
|---|---|
| 12 | 2, 3, 4, 6 |
| 123 | 1, 3 |
| 1234 | 1, 2 |
| 12345 | 1, 3, 5 |
| 123456 | 1, 2, 3, 4, 6 |
10. Use the digits 4, 5, 6 and 7 to make a number that is a multiple of 11.
How many different ways can you find to do this?
👀 Show answer
Try: 7456 → 7−4+5−6 = 2 → ✖️
Try: 5476 → 5−4+7−6 = 2 → ✖️
Try: 6547 → 6−5+4−7 = -2 → ✖️
Try: 5742 → 5−7+4−2 = 0 → ✔️
One possible number: 5742
You can test all 24 permutations. Answers may vary based on pattern checking.
11a. Show that 2521 is not divisible by any integer between 1 and 12.
👀 Show answer
So 2521 has no factors between 2 and 12. ✔️
11b. Rearrange the digits of 2251 to make a number divisible by 5.
👀 Show answer
Possible answer: 1525
11c. Rearrange the digits of 2251 to make a number divisible by 4.
👀 Show answer
Try: 1252 → last two = 52 ✖️
Try: 1224 → last two = 24 ✔️
Valid example: 1224
11d. Rearrange the digits of 2251 to make a number divisible by 8.
👀 Show answer
Try: 512 → ✔️
Possible arrangement: 2512
11e. Find the smallest integer larger than 2521 that is divisible by 6.
👀 Show answer
Answer: 2526
11f. Find the smallest integer larger than 2521 that is divisible by 11.
👀 Show answer
Answer: 2530
12a. Explain why any positive integer where every digit is 4 must be divisible by 2 and by 4.
👀 Show answer
Last 2 digits = 44 → divisible by 4
So all such numbers are divisible by 2 and 4 ✔️
12b. Every digit is 4. It is divisible by 5. Explain why this is impossible.
👀 Show answer
But all digits are 4 → cannot end in 0 or 5 → not divisible by 5.
12c.i. Every digit is 4. It is divisible by 3. Find a number with both these properties.
👀 Show answer
12c.ii. Is there more than one possible number? Give a reason.
👀 Show answer
Example: 444, 444444 (sum = 24) ✔️
12d.i. Every digit is 4. It is divisible by 11. Find a number with both these properties.
👀 Show answer
12d.ii. Is there more than one possible number? Give a reason for your answer.
👀 Show answer
Any number with even number of 4s will give alternating sum of 0 ✔️
🧠 Think like a Mathematician
Question: When can you tell if a number is divisible by a product like 8, 10 or 15? What rules help you decide?
Equipment: Pencil, paper
- a)$2 \times 4 = 8$
Look at this statement:
A number is divisible by 8 when it is divisible by 2 and by 4.
Do you think the statement is correct? Give evidence to justify your answer. - b)$2 \times 5 = 10$
Look at this statement:
A number is divisible by 10 when it is divisible by 2 and by 5.
Do you think the statement is correct? Give evidence to justify your answer. - c)$3 \times 5 = 15$
Look at this statement:
A number is divisible by 15 when it is divisible by 3 and by 5.
Do you think the statement is correct? Give evidence to justify your answer.
Follow-up Questions:
👀 show answer
- a: Not always correct. A number divisible by both 2 and 4 is not necessarily divisible by 8. For example, 12 is divisible by 2 and 4, but not by 8.
- b: Yes, because 2 and 5 have no common factors. If a number is divisible by both, it will be divisible by their product 10.
- c: Yes, the statement is correct. If a number is divisible by both 3 and 5 (which are coprime), then it is divisible by 15.
- General Rule: A number is divisible by the product of two integers $a \times b$ if it is divisible by both $a$ and $b$, and$a$ and $b$ have no common factors (they are coprime).