Real Numbers is the perfect chapter to open Class 10 with — it is short, it is logical, and once two big ideas click into place, the marks almost take care of themselves. Don’t worry if primes and proofs feel intimidating right now. We are going to build everything from the ground up, one small step at a time, like a teacher sitting right beside you. Go at your own pace, and don’t move to the next part until the one you are on feels comfortable.
Your Game Plan
- Warm up with what real numbers actually are.
- Learn the Fundamental Theorem of Arithmetic — the heart of this chapter.
- Use it to find HCF and LCM quickly and confidently.
- Master the one proof that appears almost every year — proving irrationality.
- Lock it in with the practice set at the end.
1. A Quick Warm-Up: What Are Real Numbers?
Every number you have ever used sits in one of two families. Rational numbers are the ones you can write as a fraction p/q (where q is not zero) — like 5, −3, 1/2, 0.75, or 0.333… . Irrational numbers are the ones you cannot write as a neat fraction — their decimals go on forever without repeating, like $\sqrt{2}$ = 1.41421356… or $\pi$ = 3.14159… . Put both families together and you get the real numbers — quite simply, every number you can mark as a point on the number line.
Real numbers = rational numbers + irrational numbers. If you can place it somewhere on the number line, it is a real number. This chapter is really about getting to know these two families better.
2. The Fundamental Theorem of Arithmetic
Here is the big idea of the whole chapter. Every composite number can be broken down into a product of prime numbers — and, wonderfully, there is only one way to do it (apart from the order you write them in). Think of primes as the building blocks of numbers, like LEGO bricks: every number is built from primes in exactly one unique way. This is called the Fundamental Theorem of Arithmetic.
To find that prime “recipe” for a number, you keep dividing by the smallest prime that fits, again and again, until you are left with 1. Let’s do a few together.
The prime factorisation of a number is unique. No matter how you break it down, you always land on the same set of primes. That uniqueness is what makes everything else in this chapter work.
1 is not a prime number, so it never appears in a prime factorisation. And always write repeated primes as powers (22, not 2 × 2) — it makes the next step, HCF and LCM, far easier.
3. Finding HCF and LCM the Smart Way
Once you have the prime factorisations, HCF and LCM become almost mechanical. Here are the two rules — learn them by heart:
HCF = multiply the smallest power of each prime that is common to all the numbers.
LCM = multiply the greatest power of every prime that appears in any of the numbers.
Common prime is only 2; its smallest power is 22, so HCF = 22 = 4.
For LCM, take the greatest power of every prime: 25 × 3 × 101 = LCM = 9696.
Common to all three: 2 (smallest power 21) and 3 (smallest power 31) → HCF = 2 × 3 = 6.
Greatest power of every prime: 23 × 32 × 5 = LCM = 360.
There is also a lovely shortcut that works for exactly two numbers:
For any two numbers a and b: HCF × LCM = a × b. So if you know three of these four values, you can always find the fourth.
Other number $= (HCF \times LCM) \div (\text{one number}) = (9 \times 90) \div 18 = 810 \div 18 =$ 45.
They meet again after LCM(12, 15) = 60 minutes. So the next time is 10:00 am.
The shortcut HCF × LCM = product works only for two numbers. For three or more, always go back to the prime-factorisation rules.
Whenever a question mixes “earliest time they meet again”, “largest tile that fits exactly”, or “maximum students per row”, it is secretly an LCM or HCF question. “Largest / maximum that divides” → HCF. “Earliest / next time / smallest common” → LCM.
4. Proving a Number Is Irrational
This is the part students fear the most — and it is actually the most predictable, because the same proof appears almost every year. We use a clever, backwards method called proof by contradiction: we pretend the number is rational, follow the logic honestly, and watch it crash into something impossible. Since our pretence breaks mathematics, the number cannot be rational — so it must be irrational. Let’s walk through the classic proof slowly.
Step 2. Square both sides: 2 = p2/q2, so p2 = 2q2. This means p2 is even, and if p2 is even then p itself is even. So write p = 2m.
Step 3. Substitute: (2m)2 = 2q2 → 4m2 = 2q2 → q2 = 2m2. So q2 is even, which means q is even too.
Step 4. But now both p and q are even — they share a common factor of 2. That contradicts Step 1, where we said they had no common factor. The pretence has broken. Therefore $\sqrt{2}$ cannot be rational, so $\sqrt{2}$ is irrational.
The beautiful thing is that $\sqrt{3}$ and $\sqrt{5}$ are proved in exactly the same four steps — just replace 2 with 3 or 5. Learn the flow once and you own all of them.
You must state at the start that p and q have no common factor (are coprime / in lowest terms). The entire contradiction depends on it — leave it out and the proof earns no marks.
This proof is near-guaranteed in the board paper. Memorise the flow — assume rational and coprime → square → show p even → show q even → contradiction — not the exact words. Then you can reproduce it for $\sqrt{2}$, $\sqrt{3}$ or $\sqrt{5}$ under any pressure.
Practice Worksheet
Try each question fully on paper first, then tap Show Answer to check yourself.
Q1. Express 3825 as a product of its prime factors.
Show Answer
Q2. Find the HCF and LCM of 90 and 144 by prime factorisation, and verify HCF × LCM = product of the numbers.
Show Answer
Q3. The HCF of two numbers is 12 and their LCM is 240. If one number is 48, find the other.
Show Answer
Q4. Two bells toll at intervals of 9 and 12 minutes. If they toll together at 6:00 am, when will they next toll together?
Show Answer
Q5. Prove that $\sqrt{3}$ is irrational.
Show Answer
Q6. Prove that 5 − $\sqrt{3}$ is irrational.
Show Answer
Once these feel easy, you have genuinely mastered Real Numbers. Don’t aim for perfect on the first try — aim for one more correct question than yesterday.
1. What is the HCF of 96 and 404?
2. What is the LCM of 6, 72 and 120?
3. Which of these is an irrational number?
4. HCF of two numbers is 9 and their LCM is 90. If one number is 18, the other is:
5. The Fundamental Theorem of Arithmetic says the prime factorisation of a composite number is:
6. 3825 written as a product of primes is: