Class 10th Maths Ch 1 Solutions

You stare at the textbook. Exercise 1.1, Question 2. "Use Euclid’s division algorithm to find the HCF of 135 and 225.

It looks simple enough. Two numbers. Which means been there. A few steps. Then you get stuck on the remainder step, or you forget which number goes where in the lemma statement, and suddenly the "easy chapter" feels like a wall. Most of us have.

Class 10 Maths Chapter 1 — Real Numbers* — has a reputation. It’s short. It looks straightforward. And because it looks straightforward, students tend to rush it. In practice, that’s the trap. Consider this: this chapter isn't just about finding HCF and LCM; it’s the bedrock of number theory for everything that comes after — polynomials, coordinate geometry, even calculus later on. If the foundation cracks here, the whole building wobbles.

Let’s walk through it properly. No fluff. Just the concepts that actually matter, the traps that catch everyone, and how to solve these problems without losing your mind.

What Is Class 10 Maths Chapter 1 Actually About?

Strip away the exercise numbers and the board exam pressure. At its core, Real Numbers is about understanding what numbers are and how they behave.

The NCERT textbook divides the chapter into four main exercises, each targeting a specific pillar:

1. Euclid’s Division Lemma & Algorithm (Ex 1.1) — The mechanics of divisibility. Finding HCF. Word problems that disguise the same logic.
2. Fundamental Theorem of Arithmetic (Ex 1.2) — Prime factorization. The idea that every composite number breaks down into primes in exactly one way. HCF and LCM using prime factors. The famous HCF × LCM = Product of Numbers relationship (and when it doesn't* apply for three numbers).
3. Revisiting Irrational Numbers (Ex 1.3) — Proof by contradiction. Proving √2, √3, √5, or expressions like 2 + √3 are irrational. This is pure logic, not calculation.
4. Rational Numbers & Decimal Expansions (Ex 1.4) — The terminating vs. non-terminating repeating decision rule. Looking at the denominator’s prime factors (only 2 and 5? Terminating. Anything else? Non-terminating repeating).

That’s the map. Four exercises. Four distinct skill sets. Miss one, and the others get shaky.

Why This Chapter Matters More Than You Think

"Sir, this is just HCF and LCM. We did this in Class 6."

Yes. And no.

In Class 6, you memorized steps. Now, in Class 10, you prove why the steps work. The board exam — and frankly, JEE, NTSE, or any competitive test down the line — doesn't ask "Find HCF of 12 and 18.Now, " It asks: "Three alarm clocks ring at intervals of 4, 12, and 20 minutes. If they start together, after how long will they ring together again?" That’s LCM wearing a disguise.

Or: "Prove that √5 is irrational.Because of that, " You can't guess that. You need the structure of contradiction: Assume it's rational → express as p/q in simplest form → show both p and q share a factor → contradiction.

This chapter teaches mathematical rigor. It’s the first time many students encounter a formal proof that isn't geometry. That skill — logical deduction — transfers to every other chapter. Consider this: polynomials? That said, you need the Factor Theorem, which relies on division algorithms. So quadratic equations? The discriminant logic echoes the "nature of roots" thinking you build here.

Skip the depth here, and you’re memorizing formulas in Chapter 2 without knowing why they work.

How to Actually Solve These Problems (Concept by Concept)

### Euclid’s Division Lemma vs. Algorithm — Know the Difference

This is where marks vanish silently.

The Lemma is a statement: Given positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.* It’s an existence theorem. You state* it. You apply* it to prove things (like "square of any positive integer is of form 3m or 3m+1").

The Algorithm is the process* of finding HCF. You apply the lemma repeatedly until remainder is zero. The divisor at that stage is the HCF.

In practice:

• If the question says "Use Euclid’s division algorithm to find HCF..." → Do the long division steps. Show every line: 225 = 135 × 1 + 90, 135 = 90 × 1 + 45, 90 = 45 × 2 + 0. HCF = 45.
• If it says "Using Euclid’s division lemma, show that..." → You’re doing algebra with a = bq + r. You plug in b = 3 (or 6, or whatever the question demands) and analyze the remainders r = 0, 1, 2...

Don't mix them up. Examiners check for the method* keyword.

Want to learn more? We recommend lithospheric plates can consist of which of the following components and what is the domain of the relation for further reading.

### Fundamental Theorem of Arithmetic — Prime Factorization is King

Exercise 1.2 lives and dies by the factor tree.

The rule: Every composite number = product of primes, uniquely (order doesn't matter).

HCF via Prime Factors: Take the lowest* power of common* primes. LCM via Prime Factors: Take the highest* power of all primes present.

Example: 144 and 198.144 = 2⁴ × 3² 198 = 2 × 3² × 11 HCF = 2¹ × 3² = 18 LCM =

2² × 3² × 11 = 396

Irrational Numbers — The Contradiction Toolkit

Here's the template every proof follows:

1. Assume the opposite — suppose √5 = p/q where p, q are coprime integers
2. Square both sides — 5 = p²/q² → 5q² = p²
3. Show divisibility — p² is divisible by 5 → p is divisible by 5
4. Substitute and repeat — let p = 5k → 5q² = 25k² → q² = 5k² → q is also divisible by 5
5. Contradict coprimality — both p and q share factor 5 → contradiction

This isn't just about √5. The same logic proves √3, √17, or 3+√2 are irrational.

Decimal Expansions — Terminating vs. Non-terminating

The golden rule: A rational number p/q (in simplest form) has a terminating decimal if and only if the denominator q is of the form 2ⁿ5ᵐ where n, m are non-negative integers.

Examples:

• 7/40 = 7/(2³×5) → terminates
• 11/72 = 11/(2³×3²) → doesn't terminate (has factor 3)
• 13/125 = 13/5³ → terminates

Why This Chapter Deserves Your Full Attention

Real talk — Chapter 1 isn't just warm-up. It's the foundation that determines how smoothly you glide through:

• Polynomials (Factor Theorem relies on remainder concepts)
• Quadratic Equations (discriminant analysis mirrors proof logic)
• Coordinate Geometry (distance formula applications)
• Trigonometry (proof-based identities)
• Calculus (limit definitions, continuity proofs)

Students who rush through this chapter often hit a wall in later topics because they lack the logical scaffolding. They memorize procedures instead of understanding structures.

The Bottom Line: Master this chapter not because it's easy, but because it teaches you how to think mathematically. Every problem here builds the same muscle — precise reasoning, structured argumentation, and pattern recognition — that you'll need for every advanced topic.

Don't just solve these problems. Worth adding: understand why each step works. Because when you face that twisted LCM problem in the board exam or need to prove irrationality under time pressure, it won't be formulas that save you — it'll be the clarity of thought you build right here.

That's the real payoff of Real Numbers.