Why Real Numbers Matter Beyond the Exam: HCF and LCM in Everyday Life

Students often meet HCF and LCM as two lists of steps to memorise, and then forget which is which at the exam. The way out is to see them as answers to two kinds of everyday questions. One is about cutting or sharing things into equal parts. The other is about when two repeating events line up again. Once you can tell which question you are facing, you will rarely pick the wrong one.

This guide builds the idea through tiles, buses, traffic signals and runners, then goes under the surface to prime factorisation, and finishes with decimals and irrational numbers. For the full chapter, keep these real numbers notes and the chapter notes for Class 10 nearby, and for more everyday uses, see real numbers and their applications.

Two Questions, Two Tools

A simple rule of thumb is that HCF is a divisor, so it is never larger than the smallest number, and LCM is a multiple, so it is never smaller than the largest number. If your answer breaks that rule, you have swapped them.

Cutting and Sharing: Where HCF Lives

Tiling a floor

A hall is 720 cm long and 540 cm wide. What is the side of the largest square tile that fits exactly, and how many tiles are needed?

The tile side must divide both 720 and 540, and the largest such number is the HCF. By Euclid’s method, 720 = 540 × 1 + 180, and 540 = 180 × 3 + 0, so the HCF is 180 cm. The number of tiles is (720 ÷ 180) × (540 ÷ 180) = 4 × 3 = 12 tiles.

Packing identical gift bags

A shop has 144 pencils and 96 erasers. What is the largest number of identical gift bags it can make, using all of them? The HCF of 144 and 96 is 48, so there can be 48 bags, each with 144 ÷ 48 = 3 pencils and 96 ÷ 48 = 2 erasers.

The same idea explains why a carpenter cuts planks of different lengths into the longest equal pieces with no waste, and why a teacher arranges students from two classes into the greatest number of identical teams.

Meeting Again: Where LCM Lives

Two buses at a depot

Two buses leave a depot at 6:00 a.m. One returns every 20 minutes and the other every 30 minutes. When will they next leave together? The LCM of 20 and 30 is 60, so they leave together again at 7:00 a.m.

Three traffic signals

Three traffic lights change every 48, 72 and 108 seconds. If they change together now, when will they change together again?

Factorise: 48 = 2⁴ × 3, 72 = 2³ × 3², 108 = 2² × 3³. Take the highest power of each prime: LCM = 2⁴ × 3³ = 16 × 27 = 432 seconds, which is 7 minutes 12 seconds.

Two runners on a circular track

Two runners start together and run in the same direction. One takes 18 minutes to complete a lap and the other takes 12. They meet again at the starting point after the LCM of 18 and 12, which is 36 minutes.

The same logic fits temple bells that ring at different intervals, repeating patterns in a rangoli or border design, and the way two gears with different numbers of teeth return to the same starting position together.

Prime Factorisation: The Engine Under Both

The Fundamental Theorem of Arithmetic says that every composite number can be written as a product of primes in exactly one way, apart from the order of the factors. That uniqueness is what makes the following method work for every pair of numbers.

• HCF: take each prime that appears in both numbers, with the smallest power.
• LCM: take every prime that appears in either number, with the largest power.

Example. Find the HCF and LCM of 96 and 404.

96 = 2⁵ × 3 and 404 = 2² × 101.

• HCF: the only common prime is 2, with the smallest power 2². HCF = 4.
• LCM: 2⁵ × 3 × 101 = 32 × 303 = 9696.

Check with the relation that holds for two numbers: HCF × LCM = product of the numbers. Here 4 × 9696 = 38,784, and 96 × 404 = 38,784. ✓

This relation does not work for three or more numbers, and it is a classic trap.

Decimals That End and Decimals That Repeat

Every rational number can be written as a decimal, and the decimal either ends or repeats. Prime factorisation tells you which, before you divide.

Write the fraction in its lowest terms. If the denominator has only 2 and 5 as prime factors, the decimal terminates. If it has any other prime factor, the decimal is non-terminating and repeating.

Irrational Numbers: Why √2 Cannot Be a Fraction

A number that is not rational is irrational, and its decimal never ends and never repeats. The classic example is √2, and the proof is a short argument by contradiction. Read the ideas first in these notes on revisiting irrational numbers.

1. Assume √2 = a/b, where a and b are whole numbers with no common factor other than 1.
2. Square both sides: 2 = a²/b², so a² = 2b². Therefore a² is even, and so a must be even. Write a = 2c.
3. Substitute: 4c² = 2b², so b² = 2c². Therefore b² is even, and so b is even.
4. Contradiction: both a and b are even, so they share the factor 2, which contradicts the assumption that they had no common factor.

So √2 cannot be written as a/b. It is irrational. The same pattern proves that √3, √5 and the square roots of other primes are irrational. Be careful with questions that add a rational and an irrational number: the sum is always irrational.

Where Real Numbers Show Up Next

The chapter is not only about numbers on paper. The uniqueness of prime factorisation is the basis of modern security, because multiplying two large primes is easy but recovering them from the product is very hard, and that difficulty protects data online. The idea of repeating cycles, which LCM captures, appears in calendars, timetables and signals. Understanding why a method works makes you better at every chapter that follows, including the numerical types gathered in the 15 maths numerical types that appear every year.

Slips That Cost Marks

Real Numbers in Your Textbooks

Real Numbers is the first chapter of the Class 10 Maths book in both the CBSE and the Karnataka SSLC textbooks. Revise with the chapter MCQs and these important questions with previous-year problems. Published deleted-syllabus lists differ on how much of Euclid’s division algorithm CBSE still expects, so check your own board’s current syllabus for that sub-topic. The ideas return in Class 11 and feed into number theory questions in JEE Main and KCET.

Frequently Asked Questions

Q1. When should I use HCF and when LCM?

Use HCF when you are cutting or sharing into the largest equal parts, and use LCM when you are finding when repeating events happen together again.

Q2. What is the Fundamental Theorem of Arithmetic?

Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the factors.

Q3. How do I know if a fraction gives a terminating decimal?

Write it in lowest terms. If the denominator’s only prime factors are 2 and 5, the decimal terminates. Otherwise it is non-terminating and repeating.

Q4. Is HCF × LCM = product of the numbers true for three numbers?

No. The relation holds only for two numbers.

Q5. Why is √2 irrational?

If √2 were a/b in lowest terms, then a and b would both turn out to be even, which contradicts having no common factor. So √2 cannot be written as a fraction.

Numbers With a Purpose

HCF and LCM are not two lists of steps. They are two answers to two everyday questions: what is the biggest piece that fits, and when do we line up again. Ask that first, factorise second, and the chapter becomes one of the easiest in the book.