How to Tell If a Number Divides Another Without Dividing
Learn the divisibility tests, how to split a number into prime factors, and how to find HCF and LCM so word problems about tiles and bells become straightforward.
What are factors and multiples?
A factor of a number divides it exactly, leaving no remainder. A multiple is what you get when you multiply a number by a whole number. Since , both 3 and 4 are factors of 12, and 12 is a multiple of each of them.
This page covers everything in the ICSE Class 6 Mathematics chapter on factors, multiples, HCF and LCM: how numbers are classified, the divisibility tests that save you long division, prime factorisation, and the two methods for finding HCF and LCM.
This page covers everything in the ICSE Class 6 Mathematics chapter on factors, multiples, HCF and LCM: how numbers are classified, the divisibility tests that save you long division, prime factorisation, and the two methods for finding HCF and LCM.
How are numbers classified as prime, composite or co-prime?
A number is even if 2 divides it and odd if it does not.
A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13 and so on. A composite number has more than two factors, such as 12, whose factors are 1, 2, 3, 4, 6 and 12.
1 is neither prime nor composite, because it has only one factor. And 2 is the only even prime — every other even number has 2 as an extra factor, which disqualifies it.
Co-prime numbers are two numbers whose only common factor is 1, such as 8 and 15. They need not be prime themselves. Twin primes are primes differing by 2, such as 11 and 13. A perfect number equals the sum of its factors excluding itself: the factors of 6 apart from 6 are 1, 2 and 3, and .
For example, when sharing 15 sweets among 8 children you cannot give a whole number each, precisely because 8 and 15 are co-prime.
A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13 and so on. A composite number has more than two factors, such as 12, whose factors are 1, 2, 3, 4, 6 and 12.
1 is neither prime nor composite, because it has only one factor. And 2 is the only even prime — every other even number has 2 as an extra factor, which disqualifies it.
Co-prime numbers are two numbers whose only common factor is 1, such as 8 and 15. They need not be prime themselves. Twin primes are primes differing by 2, such as 11 and 13. A perfect number equals the sum of its factors excluding itself: the factors of 6 apart from 6 are 1, 2 and 3, and .
For example, when sharing 15 sweets among 8 children you cannot give a whole number each, precisely because 8 and 15 are co-prime.
What are the divisibility tests?
Each test lets you check a factor by inspection instead of dividing.
- 2: the last digit is even.
- 3: the sum of the digits is divisible by 3.
- 4: the number formed by the last two digits is divisible by 4.
- 5: the last digit is 0 or 5.
- 6: divisible by both 2 and 3.
- 8: the last three digits form a number divisible by 8.
- 9: the digit sum is divisible by 9.
- 10: the last digit is 0.
- 11: the difference between the sum of digits in odd places and the sum in even places is 0 or a multiple of 11.
Test 4,176. Last digit 6 is even, so 2 divides it. Digit sum , which 3 and 9 both divide. Last two digits give 76, and , so 4 divides it. Being divisible by both 2 and 3, it is divisible by 6 as well.
The test for 6 is the one to be careful with: passing only the test for 2, or only the test for 3, is not enough — it must pass both.
- 2: the last digit is even.
- 3: the sum of the digits is divisible by 3.
- 4: the number formed by the last two digits is divisible by 4.
- 5: the last digit is 0 or 5.
- 6: divisible by both 2 and 3.
- 8: the last three digits form a number divisible by 8.
- 9: the digit sum is divisible by 9.
- 10: the last digit is 0.
- 11: the difference between the sum of digits in odd places and the sum in even places is 0 or a multiple of 11.
Test 4,176. Last digit 6 is even, so 2 divides it. Digit sum , which 3 and 9 both divide. Last two digits give 76, and , so 4 divides it. Being divisible by both 2 and 3, it is divisible by 6 as well.
The test for 6 is the one to be careful with: passing only the test for 2, or only the test for 3, is not enough — it must pass both.
How do you find the HCF of two numbers?
The HCF (highest common factor) is the largest number that divides all the given numbers exactly.
By prime factorisation, split each number into primes and multiply the primes they share:
The shared primes are , so .
By long division, divide the larger number by the smaller, then divide the previous divisor by the remainder, repeating until the remainder is 0. The last divisor is the HCF. For 60 and 36: leaves 24; leaves 12; leaves 0. The HCF is 12.
HCF is the tool for questions about cutting or grouping into the largest equal parts — the biggest tile that fits a floor exactly, or the largest number of identical gift bags you can fill.
By prime factorisation, split each number into primes and multiply the primes they share:
The shared primes are , so .
By long division, divide the larger number by the smaller, then divide the previous divisor by the remainder, repeating until the remainder is 0. The last divisor is the HCF. For 60 and 36: leaves 24; leaves 12; leaves 0. The HCF is 12.
HCF is the tool for questions about cutting or grouping into the largest equal parts — the biggest tile that fits a floor exactly, or the largest number of identical gift bags you can fill.
Formula
How do you find the LCM, and how is it linked to the HCF?
The LCM (lowest common multiple) is the smallest number that all the given numbers divide exactly. By prime factorisation, take every prime that appears, each to the highest power it reaches in any of the numbers:
For two numbers there is a relation worth memorising, because it turns one quantity into the other:
Check it with our pair. Here and , so . And as well, so the relation holds.
This gives a shortcut. If two numbers are 18 and 24 with HCF 6, then
Note that this relation works for two numbers only — it does not extend to three.
For two numbers there is a relation worth memorising, because it turns one quantity into the other:
Check it with our pair. Here and , so . And as well, so the relation holds.
This gives a shortcut. If two numbers are 18 and 24 with HCF 6, then
Note that this relation works for two numbers only — it does not extend to three.
Exam tip
Exam tip: knowing whether a word problem wants HCF or LCM
Almost every mark lost here comes from choosing the wrong one of the two.
Ask what the answer describes. If it is the size of each equal part, or the largest possible group, it is HCF — cutting ribbon into the longest equal pieces, or the biggest square tile that paves a room.
If it is a moment when things coincide again, or the smallest quantity that several numbers divide, it is LCM — bells ringing together, buses leaving at the same time, or the least number of sweets that can be shared equally among different-sized groups.
The shortcut: "largest, cut, share out" points to HCF; "smallest, together again, meet" points to LCM.
Ask what the answer describes. If it is the size of each equal part, or the largest possible group, it is HCF — cutting ribbon into the longest equal pieces, or the biggest square tile that paves a room.
If it is a moment when things coincide again, or the smallest quantity that several numbers divide, it is LCM — bells ringing together, buses leaving at the same time, or the least number of sweets that can be shared equally among different-sized groups.
The shortcut: "largest, cut, share out" points to HCF; "smallest, together again, meet" points to LCM.
Did you know
Why is 1 not counted as a prime number?
A prime is defined as having exactly two distinct factors. The number 1 has only one factor — itself — so it fails the definition.
There is a practical reason too. Prime factorisation is useful because each number breaks into primes in exactly one way. If 1 were prime, you could insert it endlessly — or or — and that uniqueness would be lost.
There is a practical reason too. Prime factorisation is useful because each number breaks into primes in exactly one way. If 1 were prime, you could insert it endlessly — or or — and that uniqueness would be lost.
Key takeaways
Factors, HCF and LCM in 30 seconds
- A factor divides a number exactly; a multiple is the result of multiplying by a whole number.
- Primes have exactly two factors, composites have more, 1 is neither, and 2 is the only even prime; co-prime numbers share no factor except 1.
- Divisibility tests check factors by inspection; for 6 the number must pass the tests for both 2 and 3.
- HCF is the largest common factor, found by prime factorisation or repeated long division, and answers "largest equal part" questions.
- LCM is the smallest common multiple and answers "when do they coincide" questions; for two numbers, HCF × LCM equals their product.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Primes have exactly two factors, composites have more, 1 is neither, and 2 is the only even prime; co-prime numbers share no factor except 1.
- Divisibility tests check factors by inspection; for 6 the number must pass the tests for both 2 and 3.
- HCF is the largest common factor, found by prime factorisation or repeated long division, and answers "largest equal part" questions.
- LCM is the smallest common multiple and answers "when do they coincide" questions; for two numbers, HCF × LCM equals their product.
You will remember all of this far better after answering five questions on it than after reading it twice.