The Biggest Square Tile That Fits a Floor With No Cutting
Learn prime factorisation by factor tree and division, find the HCF and LCM from the factors, decide which one a word problem needs, and use the HCF times LCM relationship.
What is the largest square tile that can cover a floor with no cutting?
It is the HCF of the floor's two side lengths. For a room 36 cm by 48 cm in a scale drawing, the answer is a 12 cm tile — and prime factors tell you that in one line, with no trial and error.
This page covers everything in the CBSE Class 7 Mathematics chapter on common factors and multiples: prime factorisation by factor tree and by division, finding the HCF and the LCM, deciding which one a word problem wants, and the relationship that links them.
This page covers everything in the CBSE Class 7 Mathematics chapter on common factors and multiples: prime factorisation by factor tree and by division, finding the HCF and the LCM, deciding which one a word problem wants, and the relationship that links them.
How do you write a number as a product of prime factors?
Keep breaking the number into factors until every piece is a prime — a number with exactly two factors, itself and 1.
The factor tree method splits as you go. For 36: split into , then 4 into and 9 into . So
The division method divides repeatedly by the smallest prime that fits. For 48: divide by 2 to get 24, by 2 to get 12, by 2 to get 6, by 2 to get 3, then by 3 to get 1. So
Both methods reach the same answer, which is the point: a number has exactly one set of prime factors, however you split it. Starting 36 as instead of still ends at .
One detail catches students out: 1 is not a prime, because it has only one factor. So it never appears in a prime factorisation.
The factor tree method splits as you go. For 36: split into , then 4 into and 9 into . So
The division method divides repeatedly by the smallest prime that fits. For 48: divide by 2 to get 24, by 2 to get 12, by 2 to get 6, by 2 to get 3, then by 3 to get 1. So
Both methods reach the same answer, which is the point: a number has exactly one set of prime factors, however you split it. Starting 36 as instead of still ends at .
One detail catches students out: 1 is not a prime, because it has only one factor. So it never appears in a prime factorisation.
How do you find the HCF and what does it tell you?
The HCF — highest common factor — is the largest number that divides both. From prime factors, take every prime the two numbers share, each to the lower power.
For 36 and 48:
Both contain 2 and 3. The lower powers are and , so
The tile problem now answers itself. A 36 cm by 48 cm floor takes 12 cm square tiles, because 12 divides both sides exactly, and tiles cover it.
HCF questions always ask for the largest of something being split or shared — the biggest identical packets, the longest equal pieces of ribbon, the greatest number of children who can share sweets with none left over.
If two numbers share no prime factor, their HCF is 1 and they are called co-prime. So 8 and 15 are co-prime, even though neither is itself a prime.
For 36 and 48:
Both contain 2 and 3. The lower powers are and , so
The tile problem now answers itself. A 36 cm by 48 cm floor takes 12 cm square tiles, because 12 divides both sides exactly, and tiles cover it.
HCF questions always ask for the largest of something being split or shared — the biggest identical packets, the longest equal pieces of ribbon, the greatest number of children who can share sweets with none left over.
If two numbers share no prime factor, their HCF is 1 and they are called co-prime. So 8 and 15 are co-prime, even though neither is itself a prime.
How do you find the LCM and when do you need it?
The LCM — lowest common multiple — is the smallest number that both divide into. From prime factors, take every prime that appears in either number, each to the higher power.
For 36 and 48, the primes are 2 and 3, with higher powers and :
LCM questions are about things repeating and coinciding. Two buses leave a stand every 12 minutes and every 18 minutes. Since and , the LCM is , so they leave together every 36 minutes.
The sorting rule is worth memorising, because choosing the wrong one is the main source of lost marks here. splitting into equal parts needs the HCF, and events coming together again need the LCM.
A size check also helps: the HCF is never larger than the smaller number, and the LCM is never smaller than the larger number. An "HCF" of 144 for 36 and 48 is impossible on sight.
For 36 and 48, the primes are 2 and 3, with higher powers and :
LCM questions are about things repeating and coinciding. Two buses leave a stand every 12 minutes and every 18 minutes. Since and , the LCM is , so they leave together every 36 minutes.
The sorting rule is worth memorising, because choosing the wrong one is the main source of lost marks here. splitting into equal parts needs the HCF, and events coming together again need the LCM.
A size check also helps: the HCF is never larger than the smaller number, and the LCM is never smaller than the larger number. An "HCF" of 144 for 36 and 48 is impossible on sight.
Formula
Why does HCF times LCM equal the product of the two numbers?
For any two numbers and :
Check it on 36 and 48:
The reason is in the prime powers. For each prime, the HCF takes the lower power and the LCM takes the higher one, so between them they use both powers exactly once — which is what multiplying by does.
The formula is a shortcut when three of the four values are known. If two numbers multiply to 1728 and their HCF is 12, then
with no factorising needed.
Use it as a check on every HCF-and-LCM question: if your two answers multiplied together do not match the product of the original numbers, one of them is wrong. Note the limit, though — the relationship holds for two numbers only, so it cannot be applied to a set of three.
Check it on 36 and 48:
The reason is in the prime powers. For each prime, the HCF takes the lower power and the LCM takes the higher one, so between them they use both powers exactly once — which is what multiplying by does.
The formula is a shortcut when three of the four values are known. If two numbers multiply to 1728 and their HCF is 12, then
with no factorising needed.
Use it as a check on every HCF-and-LCM question: if your two answers multiplied together do not match the product of the original numbers, one of them is wrong. Note the limit, though — the relationship holds for two numbers only, so it cannot be applied to a set of three.
Exam tip
Exam tip: telling an HCF question from an LCM question
Before factorising anything, decide which quantity the question wants, because the arithmetic is short and the choice is where the marks are.
Look for the giveaway words. Largest, greatest, maximum, equal groups, divides exactly point to the HCF. Least, smallest, again together, at the same time, minimum number of point to the LCM.
Then apply the size check before writing your final line: an HCF must be at most the smaller number, and an LCM at least the larger.
Lay out both factorisations in index form, one under the other, so the shared primes line up in columns. Reading off lower powers for the HCF and higher powers for the LCM then becomes a glance rather than a hunt.
Look for the giveaway words. Largest, greatest, maximum, equal groups, divides exactly point to the HCF. Least, smallest, again together, at the same time, minimum number of point to the LCM.
Then apply the size check before writing your final line: an HCF must be at most the smaller number, and an LCM at least the larger.
Lay out both factorisations in index form, one under the other, so the shared primes line up in columns. Reading off lower powers for the HCF and higher powers for the LCM then becomes a glance rather than a hunt.
Did you know
Why does every number have only one set of prime factors?
Because a prime cannot be broken down any further, so once every piece is prime there is nowhere left to go.
Start 48 as and you reach . Start it as and you reach . Same five primes, different order.
That uniqueness is what makes prime factors so useful for HCF and LCM: the factor list is a fixed fingerprint of the number, so comparing two lists compares the numbers themselves.
Start 48 as and you reach . Start it as and you reach . Same five primes, different order.
That uniqueness is what makes prime factors so useful for HCF and LCM: the factor list is a fixed fingerprint of the number, so comparing two lists compares the numbers themselves.
Key takeaways
HCF and LCM: quick revision
- Prime factorisation by factor tree or repeated division gives the same unique answer; 1 is not a prime.
- HCF takes the shared primes to the lower power: for and , the HCF is .
- LCM takes all primes to the higher power, giving .
- Splitting into the largest equal parts needs the HCF; events repeating and coinciding need the LCM.
- for two numbers, so — use it to check both answers at once.
- The HCF cannot exceed the smaller number and the LCM cannot be below the larger one.
You will remember all of this far better after answering five questions on it than after reading it twice.
- HCF takes the shared primes to the lower power: for and , the HCF is .
- LCM takes all primes to the higher power, giving .
- Splitting into the largest equal parts needs the HCF; events repeating and coinciding need the LCM.
- for two numbers, so — use it to check both answers at once.
- The HCF cannot exceed the smaller number and the LCM cannot be below the larger one.
You will remember all of this far better after answering five questions on it than after reading it twice.