LCM and HCF Made Easy: The Two Little Ideas Behind Tiles, Timetables and Tricky Exam Questions A Student Who Guessed His Way Through Maths...
LCM and HCF Made Easy: The Two Little Ideas Behind Tiles, Timetables and Tricky Exam Questions
A
few years ago, a Class 6 student I will call Arjun stayed back after my maths
period. He opened his notebook, pointed at two pages of neat working, and said,
"Sir, I can find LCM. I can find HCF. But when a word problem comes, I
just guess which one to use. Fifty-fifty chance, right?"
I
smiled, because I had seen this a hundred times. Arjun was not weak in maths.
He had learned the steps of LCM and HCF without ever learning the story behind
them.
So
I closed his notebook and asked a different question. "You have 12 pencils
and 18 erasers. You want to pack them into identical kits with nothing left
over. What is the largest number of kits you can make?" He thought for a
minute, scribbled on a scrap of paper, and said, "Six kits. Two pencils
and three erasers in each."
"Good,"
I said. "You just found the HCF without using a single formula." Then
I asked another one. "Two lights blink together right now. One blinks
every 12 seconds and the other every 18 seconds. When will they blink together
again?" He counted in twelves and eighteens until both lists met at 36
seconds. "And that," I told him, "was the LCM."
His
face changed. "So HCF is about sharing things equally, and LCM is about
things meeting again?" Exactly. Two weeks later, Arjun was the first in
the class to finish the word-problem section of his unit test, and he did not
guess even once.
If
you have ever felt like Arjun, this post is for you. We will learn LCM and HCF
in plain English, with small stories, three easy methods, a comparison table,
the mistakes to avoid, and five study habits that really work. Take a pencil,
sit comfortably, and let us begin.
LCM
and HCF are two ways of comparing numbers. Both are built on two ideas you
already know from earlier classes: factors and multiples. Before anything else,
let us make those two words crystal clear.
•
A factor of a number divides it exactly,
leaving no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12.
•
A multiple of a number is what you get when
you multiply it by 1, 2, 3 and so on. The multiples of 12 are 12, 24, 36, 48,
60 and onwards.
•
Factors are limited in number and never bigger
than the number itself. Multiples go on forever and are never smaller than the
number.
HCF
stands for Highest Common Factor. You may also see it called GCD
(Greatest Common Divisor) or GCF (Greatest Common Factor). All three names mean
the same thing: the biggest number that divides two or more numbers exactly.
Take
12 and 18. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2,
3, 6, 9, 18. The common ones are 1, 2, 3 and 6, and the highest of these is 6.
So the HCF of 12 and 18 is 6.
LCM
stands for Lowest Common Multiple, sometimes written as Least Common
Multiple. It is the smallest number (other than zero) that is a multiple of
every number you are comparing.
Look
at 12 and 18 again. Multiples of 12: 12, 24, 36, 48, 60, 72. Multiples of 18:
18, 36, 54, 72. Both lists share 36 and 72, and the lowest of these is 36. So
the LCM of 12 and 18 is 36.
Here
is the little rule I give every batch. HCF breaks things down, so its answer
is never bigger than the smallest number. LCM builds things up, so its answer
is never smaller than the biggest number. In a line: HCF ≤ smallest number
≤ biggest number ≤ LCM. If your answer breaks this rule, stop and check your
working. For 12 and 18, we got 6 and 36, and the rule holds beautifully.
Many
students treat this chapter as a small stop on the way to something bigger. In
truth, LCM and HCF are tools you will pick up again and again, in school and
outside it. Here is where they show up.
•
Fractions: To add or subtract
fractions with different denominators, you need the LCM of the denominators. To
reduce a fraction to its simplest form, you divide the top and bottom by their
HCF.
•
Ratios and algebra: Taking
out a common factor while factorising an expression is the HCF idea in
disguise.
•
Exams: From school tests to Olympiads
and many competitive exams, questions on LCM and HCF appear regularly, because
they check both calculation and clear thinking.
•
Daily life: Planning schedules,
dividing things into equal groups, laying tiles, cutting cloth or ribbon, and
working out when two repeating events will line up all use these two ideas.
•
Number sense: Working
with factors and multiples trains your mind to see how numbers are built. That
makes mental maths faster and more fun.
Even
in computer science, a method for finding the HCF (which we will meet shortly)
is one of the oldest step-by-step procedures still in use today. So when you
practise this chapter, you are learning something that has lasted over two
thousand years.
There
are three reliable methods. Learn all three, then choose the one that suits the
numbers in front of you.
Write
the factors or multiples and look for the match. Let us find the HCF and LCM of
8 and 12.
•
Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3,
4, 6, 12. The highest common one is 4, so HCF = 4.
•
Multiples of 8: 8, 16, 24, 32. Multiples of 12:
12, 24, 36. The first common one is 24, so LCM = 24.
This
method is wonderful for understanding. It becomes slow, though, when numbers
grow large. Imagine listing multiples of 156 and 234. That is why we need the
next method.
Break
each number into prime factors, which are prime numbers multiplied together.
Let us try 36 and 60.
•
36 = 2 × 2 × 3 × 3 = 2² × 3²
•
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Now
the golden rule. For HCF, pick only the primes that appear in every
number, and take the lowest power of each. Here that gives 2² × 3 = 12.
For LCM, pick every prime that appears anywhere, and take the highest
power of each. Here that gives 2² × 3² × 5 = 180.
So
the HCF of 36 and 60 is 12, and the LCM is 180. A quick check: 12 × 180 = 2160,
and 36 × 60 = 2160 as well. We will see why this works in a moment.
For
LCM, use common division. Write the numbers in a row and keep dividing
by any prime that divides at least one of them. Numbers that are not divisible
simply come down unchanged. Continue until every number becomes 1, then
multiply all the divisors. Let us find the LCM of 12, 18 and 24.
•
Divide by 2: 12, 18, 24 become 6, 9, 12.
•
Divide by 3: 6, 9, 12 become 2, 3, 4.
•
Divide by 2: 2, 3, 4 become 1, 3, 2.
•
Divide by 3: 1, 3, 2 become 1, 1, 2.
•
Divide by 2: 1, 1, 2 become 1, 1, 1.
Multiply
the divisors: 2 × 3 × 2 × 3 × 2 = 72. So the LCM of 12, 18 and 24 is 72.
For
HCF, use Euclid's method. Divide the bigger number by the smaller one.
Then divide the divisor by the remainder. Keep going until the remainder is
zero. The last divisor is the HCF. Let us find the HCF of 252 and 105.
•
252 ÷ 105 gives quotient 2 and remainder 42.
•
105 ÷ 42 gives quotient 2 and remainder 21.
•
42 ÷ 21 gives quotient 2 and remainder 0.
The
last divisor is 21, so the HCF of 252 and 105 is 21. Now, using the
relationship below, the LCM is (252 × 105) ÷ 21 = 1260. Two numbers, two
answers, and almost no effort.
•
The product rule (for two numbers only): HCF ×
LCM = first number × second number. For 12 and 18: 6 × 36 = 216 = 12 × 18.
•
Co-prime numbers: If two
numbers share no common factor except 1, their HCF is 1 and their LCM is simply
their product. For 8 and 15, HCF = 1 and LCM = 120.
•
One is a multiple of the other: The HCF
is the smaller number and the LCM is the larger one. For 6 and 24, HCF = 6 and
LCM = 24.
•
HCF always divides the LCM exactly.
Check: 6 divides 36.
•
Fractions (in simplest form): HCF of
fractions = HCF of numerators ÷ LCM of denominators. LCM of fractions = LCM of
numerators ÷ HCF of denominators. For 2/3 and 4/9, the HCF is 2/9 and the LCM
is 4/3.
This
was Arjun's big problem, so let us fix it for good. Read the question and hunt
for clue words.
•
Think HCF when you see: greatest,
largest, maximum, divide equally, cut into equal pieces, equal groups with
nothing left, biggest square tile, largest container.
•
Think LCM when you see: smallest,
least, minimum, together again, at the same time, meet again, repeat, after how
many minutes, next time.
Two
famous remainder problems also come up often. First: "Find the greatest
number that divides 245 and 1029 leaving remainder 5 in each case."
Subtract the remainder first, which gives 240 and 1024, then find their HCF.
The answer is 16. Second: "Find the smallest number which leaves
remainder 5 when divided by 12, 15 and 20." Find the LCM, which is 60,
then add the remainder. The answer is 60 + 5 = 65.
Cover
the answers with your hand, try each one, and then compare.
•
Example 1: Find the HCF and LCM of
48 and 72. Here 48 = 2⁴ × 3 and 72 = 2³ × 3². The HCF takes the lowest powers,
2³ × 3 = 24. The LCM takes the highest powers, 2⁴ × 3² = 144.
Check with the product rule: 24 × 144 = 3456, and 48 × 72 = 3456.
•
Example 2: Three bells ring every
10, 15 and 20 minutes. If they ring together at 9 a.m., when will they ring
together again? The LCM of 10, 15 and 20 is 60, so they ring together again
after 60 minutes, at 10 a.m.
•
Example 3: A tailor has two ribbons,
84 cm and 126 cm long. She wants to cut them into equal pieces with no ribbon
wasted, and she wants the pieces as long as possible. The HCF of 84 and 126 is
42, so each piece is 42 cm long. She gets 2 pieces from the first ribbon and 3
from the second, which makes 5 pieces in all.
Notice
the pattern. In every example, the first thing we did was read the question
carefully and decide whether we were splitting or waiting. The calculation only
came after that.
Meera
is a composite of several Class 7 students I have taught, and I have changed
the details to protect their privacy. Her story, however, is real in every way
that matters.
In
her first unit test on this chapter, Meera scored 9 out of 25. She told me she
"froze" whenever she saw big numbers. She was copying steps from the
textbook without understanding them, and she was afraid of long working.
We
made a simple three-week plan. In week one, she wrote the prime factors
of every number from 2 to 50 in a small table and read it aloud each morning.
In week two, she collected twenty word problems and sorted them into two
piles, an HCF pile and an LCM pile, by circling the clue words. In week
three, she did a daily timed set of five mixed questions and checked each
answer with the product rule.
One
example gave her real confidence. Her family was tiling a room 480 cm by 360 cm
and wanted the biggest possible square tiles with no cutting. She found the HCF
of 480 and 360, which is 120, so each tile would be 120 cm wide. The room needs
4 tiles along one side and 3 along the other, so 12 tiles in all. She told the
whole class about it with a big smile.
Her
next test score was 21 out of 25, and the one after that was 24. The marks were
wonderful, but what pleased me more was that she started asking, "Sir, can
I make my own problem?" That question is the sign that a student has truly
understood a topic.
The
lesson is simple. Meera did not become clever overnight. She became clear:
clear about primes, clear about clue words, and clear about checking her own
work.
•
Faster fraction work: Adding,
subtracting and simplifying fractions stops being a struggle.
•
Better exam marks: These
are scoring questions. Once the method is clear, they are among the easiest
marks in the paper.
•
Sharper number sense: You
begin to see numbers as made of smaller parts, which helps in algebra and in
mental calculation.
•
Practical problem solving:
Scheduling, packing, tiling and sharing become easy to plan.
•
More confidence: Success
in one chapter lowers fear of the next one. That confidence often spreads to
the whole subject.
Think
of these benefits as savings in a bank. Every small bit of practice you put in
today earns interest later. When you reach Class 9 and Class 10 and meet
polynomials, real numbers and number patterns, you will notice that the ideas
feel familiar. That feeling of "I have seen this before" is a quiet
advantage, and it begins with chapters like this one. Students who build a firm
base here rarely feel lost when the topics get harder.
In
fifteen years, I have marked thousands of answer sheets. The same slips appear
again and again. The good news is that every one of them has an easy fix.
Students
take the highest power for HCF or the lowest power for LCM. Fix:
Remember "HCF is Humble: common primes, lowest powers. LCM is Large: all
primes, highest powers."
When
a prime appears in only one number, students leave it out. Fix: For LCM,
tick off every prime from every number before you multiply. Nothing gets
dropped.
Some
students stop dividing while a number like 3 or 5 still remains in the row. Fix:
Keep going until the entire last row is 1, 1, 1.
HCF
× LCM = product works only for two numbers. Try it with 4, 6 and 8: the LCM is
24 and the HCF is 2, so 24 × 2 = 48, but 4 × 6 × 8 = 192. They do not match. Fix:
Write "TWO NUMBERS ONLY" on top of that formula in your notes.
This
is the Arjun problem. Fix: Underline the clue words before touching the
numbers, and ask, "Am I splitting something, or waiting for something to
repeat?"
Many
wrong answers could be caught in five seconds. Fix: Check that your HCF
divides every number, that every number divides your LCM, and that the HCF ≤
smallest number ≤ biggest number ≤ LCM rule holds.
A
word to parents and teachers reading this. Please praise the thinking, not only
the marks. When a child says, "I checked it with the product rule and it
matched," that is a habit worth celebrating loudly. Habits like that stay
with a student long after the chapter is over.
Keep
this table in your notebook. It will save you time right before an exam.
|
Point |
HCF |
LCM |
|
Full form |
Highest Common Factor |
Lowest Common Multiple |
|
Other names |
GCD, GCF |
LCD (when used for denominators) |
|
Meaning |
Biggest number that divides all the given
numbers |
Smallest number that all the given numbers
divide |
|
Size of answer |
Never bigger than the smallest number |
Never smaller than the biggest number |
|
Prime factor rule |
Common primes, lowest powers |
All primes, highest powers |
|
Clue words |
Greatest, largest, equal groups, cut equally |
Smallest, least, together again, repeat |
|
Main thought |
Splitting and sharing |
Meeting and repeating |
|
Example (12 and 18) |
6 |
36 |
|
Used in fractions for |
Simplifying to lowest terms |
Adding and subtracting with unlike denominators |
|
Real-life use |
Tiles, packing kits, cutting ribbon |
Bus timings, blinking lights, repeating events |
Prime
factorisation is the engine of this chapter. Learn the primes up to 50 (2, 3,
5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47) and revise your tables up to
20. Ten minutes on this saves you an hour later.
Listing
is good for tiny numbers, but prime factorisation works every time and also
helps you spot mistakes. Use it as your first choice and bring in the division
methods when numbers get big.
Use
two coloured pens. Circle HCF clue words in blue and LCM clue words in green.
After a week of this, the right method will jump out at you without any effort.
Whenever
you have two numbers, test your answers with HCF × LCM = product. It takes ten
seconds and catches most careless errors. It also builds trust in your own
working, and trust builds speed.
Short
daily practice beats a long session once a week. Do five mixed questions every
day. Then, once a week, explain one problem to a friend, a younger sibling or
even your pet. If you can teach it, you truly know it. Better still, write one
word problem of your own and swap it with a classmate.
Try
these on a sheet of paper before reading the answers. Use any method you like.
•
Find the HCF and LCM of 20 and 30.
•
Find the LCM of 4, 6 and 10.
•
What is the side of the largest square tile that
can exactly cover a floor 150 cm by 210 cm?
•
The HCF of two numbers is 4 and their LCM is 48.
If one number is 12, what is the other?
•
What is the smallest number that is exactly
divisible by 6, 8 and 12?
Answers: (1) HCF
10 and LCM 60. (2) 60. (3) 30 cm, because the HCF of 150 and 210 is 30. (4) 16,
because 4 × 48 ÷ 12 = 16. (5) 24. If you got four or more right, you are in
great shape. If not, go back to the clue words and the prime factor method, and
try again tomorrow.
Let
us return to Arjun for a moment. He did not need a new formula. He needed a
picture in his head: HCF is about splitting equally, and LCM is about
meeting again. Once that picture was there, the steps made sense, the clue
words made sense, and the marks followed.
You
can do the same. Learn the three methods, build the habit of checking, avoid
the six common slips, and practise a little every day. LCM and HCF will soon
feel like old friends instead of exam enemies.
So
tonight, pick two numbers, any two, and find their HCF and LCM. Then ask
yourself where you might see them in real life. I would love to hear the best
example you come up with. Keep practising, stay curious, and remember that
every expert in maths once stood exactly where you are standing now.
Q1. What
does HCF stand for?
HCF
stands for Highest Common Factor. It is the largest number that divides all the
given numbers exactly.
Q2. What
does LCM stand for?
LCM
stands for Lowest Common Multiple. It is the smallest number that is a multiple
of all the given numbers.
Q3. Are HCF,
GCD and GCF the same thing?
Yes,
they are three names for the same idea. Different books and countries simply
use different names.
Q4. What is
the easiest way to find the HCF?
For
small numbers, list the factors and pick the highest common one. For bigger
numbers, use prime factorisation or Euclid's division method.
Q5. What is
the easiest way to find the LCM?
For
small numbers, list multiples until you find the first match. For bigger
numbers, use prime factorisation or the common division method.
Q6. What is
the formula connecting LCM and HCF?
For
two numbers, HCF × LCM = product of the two numbers. You can use it to find a
missing LCM or HCF quickly.
Q7. Does the
product rule work for three numbers?
No,
it works only for two numbers. For three or more, you must find the LCM and HCF
directly.
Q8. Can the
HCF be bigger than the smaller number?
No,
it cannot. The HCF must divide the smaller number, so it is always equal to or
less than it.
Q9. Can the
LCM be smaller than the bigger number?
No,
it cannot. The LCM must be a multiple of the bigger number, so it is always
equal to or more than it.
Q10. What is
the HCF of two co-prime numbers?
The
HCF of two co-prime numbers is always 1. They share no common factor other than
1.
Q11. What is
the LCM of two co-prime numbers?
It
is simply the product of the two numbers. For example, the LCM of 8 and 15 is
120.
Q12. What
are the HCF and LCM of a number with itself?
Both
are the number itself. For example, the HCF and LCM of 9 and 9 are both 9.
Q13. What
are the HCF and LCM of 1 and any number?
The
HCF is 1 and the LCM is the other number. For 1 and 14, the HCF is 1 and the
LCM is 14.
Q14. Which
one do I use to add fractions with different denominators?
Use
the LCM of the denominators. It gives the smallest common denominator and keeps
your numbers small.
Q15. Which
one do I use to simplify a fraction?
Use
the HCF of the numerator and denominator. Divide both by it to reach the lowest
terms in one step.
Q16. How do
I decide between HCF and LCM in a word problem?
Look
for clue words. Words like greatest or equal groups point to HCF, while words
like smallest or together again point to LCM.
Q17. What is
Euclid's division method?
Divide
the bigger number by the smaller one, then divide the divisor by the remainder,
and repeat. The last divisor, just before the remainder becomes zero, is the
HCF.
Q18. Why
does prime factorisation work?
Every
number breaks into primes in only one way. So comparing those primes shows
exactly what the numbers share and what each one needs.
Q19. What is
the HCF of two different prime numbers?
It
is 1, because two different primes share no factor except 1. They are always
co-prime.
Q20. What is
the LCM of two different prime numbers?
It
is their product. For example, the LCM of 5 and 7 is 35.
Q21. Can I
find LCM and HCF for zero or negative numbers?
In
school, we work with positive whole numbers. Zero and negative numbers are
handled with extra rules in higher classes.
Q22. In
which class are LCM and HCF taught?
They
are usually introduced around Classes 5 and 6. They return in Class 10 in the
chapter on Real Numbers.
Q23. How do
I find the HCF of three numbers?
Find
the HCF of the first two numbers. Then find the HCF of that answer and the
third number.
Q24. How do
I find the LCM of three numbers?
Use
prime factorisation or the common division method with all three numbers
together. You can also find the LCM of two and then take the LCM of that result
with the third.
Q25. How do
I find the HCF of fractions?
Divide
the HCF of the numerators by the LCM of the denominators. The fractions should
be in their simplest form first.
Q26. How do
I find the LCM of fractions?
Divide
the LCM of the numerators by the HCF of the denominators. Again, start with
fractions in their simplest form.
Q27. How do
I find the smallest number that leaves the same remainder when divided by
several numbers?
Find
the LCM of the divisors and add the remainder. For remainder 5 with 12, 15 and
20, the answer is 60 + 5 = 65.
Q28. How do
I find the greatest number that divides given numbers leaving remainders?
Subtract
the remainder from each number, then find the HCF of the results. For 245 and
1029 with remainder 5, find the HCF of 240 and 1024, which is 16.
Q29. How
much time should I practise every day?
Fifteen
to twenty focused minutes a day is enough for most students. Regular short
practice works better than one long weekly session.
Q30. What is
the biggest mistake students make in this chapter?
The
most common one is using the wrong tool in a word problem. Underline clue words
first and ask whether you are splitting something or waiting for it to repeat.
.webp)
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