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Real Numbers is the first chapter of Class 10 maths and usually one of the shortest. It still costs students marks in two places: HCF and LCM questions with three numbers, and proofs that a number is irrational. This post shows what is in the current syllabus, works one example of each type, lists the usual slips, and gives a routine you can finish in a week.
What is in the Class 10 Real Numbers chapter now?
CBSE's Class X Mathematics syllabus for 2026-27 lists two items under Real Numbers: the Fundamental Theorem of Arithmetic, and "proofs of irrationality of √2, √3, √5". Its explanation column adds examples like 3 + 2√5. The whole Number Systems unit is worth 6 of the 80 theory marks.
The NCERT chapter, in its rationalised 2026-27 reprint, has the same two sections: 1.2 on the Fundamental Theorem of Arithmetic and 1.3 on irrational numbers. Older guides also teach Euclid's division algorithm, but the rationalised chapter no longer has a section on it, so check your current syllabus before spending time there.
How do you find HCF and LCM by prime factorisation?
The Fundamental Theorem of Arithmetic says every composite number can be written as a product of primes in exactly one way, apart from order. That is what makes the method work.
Example 1. Find the HCF and LCM of 84 and 120.
Factorise: 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5.
HCF: take the smallest power of each common prime. Common primes are 2 and 3, so HCF = 2² × 3 = 12.
LCM: take the greatest power of every prime that appears. LCM = 2³ × 3 × 5 × 7 = 840.
Check: 12 × 840 = 10,080 and 84 × 120 = 10,080.
Example 2. Three bells ring every 12, 18 and 30 minutes. They ring together at 8:00 am. When do they next ring together?
12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5.
LCM = 2² × 3² × 5 = 180 minutes, which is 3 hours.
They ring together again at 11:00 am.
"Together again" is an LCM question. "Largest equal groups" or "largest tile that fits exactly" is an HCF question. Decide which before you calculate.
How do you prove that a number is irrational?
The proof uses contradiction: assume the number is rational, and show that leads to something impossible. The key fact, Theorem 1.2 in NCERT, is that if a prime p divides a², then p divides a.
Example 3. Prove that √3 is irrational.
Assume √3 is rational. Then √3 = a/b, where a and b are integers, b ≠ 0, and a and b are coprime (no common factor other than 1).
Square: 3 = a²/b², so a² = 3b². So 3 divides a².
Since 3 is prime, 3 divides a. Write a = 3c.
Substitute: 9c² = 3b², so b² = 3c². So 3 divides b², and therefore 3 divides b.
Now 3 divides both a and b. This contradicts step 1, where a and b were coprime.
So the assumption was wrong, and √3 is irrational.
Example 4. Prove that 3 + 2√5 is irrational, given that √5 is irrational.
Assume 3 + 2√5 = r, where r is rational.
Rearrange: √5 = (r − 3)/2.
The right side is rational, because rational numbers stay rational when you subtract and divide by non-zero integers.
That makes √5 rational, which contradicts the given fact. So 3 + 2√5 is irrational.
Why do proofs feel harder than sums?
Many students find a few numerical checks more convincing than an algebraic proof. Healy and Hoyles (2000) surveyed high-attaining 14- and 15-year-olds and found that empirical arguments dominated the proofs they built themselves, even though most knew such arguments had limits. Checking that 1.732² is close to 3 does not prove anything about √3. The chain of reasons does, which is why each line above has a "because".
Which mistakes lose marks in this chapter?
Skipping "coprime". Without it, step 5 has nothing to contradict. Write it in step 1.
Using the divisibility rule with a non-prime. 4 divides 6² = 36 but does not divide 6. The rule needs p to be prime.
Using the two-number HCF × LCM rule for three numbers. See the callout.
Taking the largest power for HCF. HCF uses the smallest power of common primes only; LCM uses the largest power of all primes.
Proving "3 + 2√5" without saying √5 is irrational. State the fact you rely on.
Stopping at the LCM. In Example 2 the question asked for a time, so 180 minutes is not the final answer.
If you keep getting stuck on the same step of a proof, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer, so you can see why each line follows from the one before.
A one-week practice routine
Day 1: Factorise ten numbers between 100 and 1000 by factor tree. Check each by multiplying back.
Day 2: Five HCF and LCM pairs, then three triples. Verify the two-number rule on the pairs only.
Day 3: Write the √2 proof from memory. Compare with NCERT line by line.
Day 4: Prove √5 and √7 irrational using the same six steps.
Day 5: Prove 5 − √3 and 2√3 + 1 irrational.
Day 6: Mixed set of six questions, timed, with HCF, LCM and proofs shuffled together.
Euclid's Elements is where the factorisation idea is first recorded, and our profile of Euclid and the idea of proof is a good read alongside this chapter. For laying out working clearly in the board paper, see common Class 10 maths board mistakes.
Today, write the √3 proof from memory in six numbered lines, then check it against Example 3.
