Real Numbers · Class 10 Mathematics: quick revision, 55 bites

Every number is built from primes

360 has a recipe, and it has only one

Just as every substance is made from elements, every whole number above 1 is a prime or a product of primes.

Takeaway. Primes are the building blocks of all whole numbers.

What makes a number prime

Two factors, no more, no fewer

Takeaway. Count the factors: exactly two means prime.

1 is not prime

The number everyone gets wrong

1 has only one factor — itself. A prime needs exactly two, so 1 is neither prime nor composite.

Takeaway. Primes start at 2.

2 is the only even prime

Even numbers are almost never prime

Every even number above 2 has 1, 2 and itself as factors — at least three — so it is composite.

Takeaway. After 2, every prime is odd.

How to grow a factor tree

Stop when every branch ends in a circle

Takeaway. A leaf that is not prime means the tree is unfinished.

The division ladder

The tidy way for big numbers

Divide by 2 until it no longer goes, then 3, then 5, then 7…

Takeaway. Smallest prime first means no factor is ever missed.

The Fundamental Theorem of Arithmetic

The theorem in one line

Takeaway. Two claims: it always exists, and there is only one.

Why unique matters

Different trees, same leaves

Takeaway. You may start a factor tree anywhere and still get the same answer.

Power form

Say it with powers

Takeaway. Examiners expect the final answer in power form.

Verify every factorisation

Check by multiplying back

Multiply your primes together. $8 \times 9 \times 5 = 360$ ✓.

Takeaway. A ten-second check saves a lost mark.

When a number ends in 0

A trailing zero needs a 2 AND a 5

Ending in 0 means divisible by $10 = 2 \times 5$.

Takeaway. Missing either prime means no trailing zero.

Why 4ⁿ never ends in 0

4, 16, 64, 256… never 0

$4^n = 2^{2n}$. Its only prime factor is 2.

Takeaway. Uniqueness is what rules the 5 out for every n.

Which powers can end in 0?

Four bases, four verdicts

Takeaway. Look for the 2 and the 5 before answering.

Proving a sum is composite

Take out what they share

$7 \times 11 \times 13 + 13 = 13 \times (77 + 1) = 13 \times 78$.

Takeaway. Factor first, add never.

Reading divisibility from a factorisation

No long division needed

$N = 2^3 \times 3^2 \times 5 \times 7$.

Takeaway. m divides N when every prime power of m fits inside N.

A quick prime test

Test only up to the square root

To test n, try primes up to $\sqrt{n}$ only.

Takeaway. If no prime up to √n divides n, then n is prime.

Primes keep money safe

Your bank password depends on it

Multiplying two huge primes is quick. Factorising the product back is so slow that online banking security relies on it.

Takeaway. Factorisation is easy to state and hard to reverse for huge numbers.

How to write a 2-mark answer

Name the theorem in your answer

For "can $6^n$ end in 0?": give the factorisation $2^n \times 3^n$, say that by the Fundamental Theorem of Arithmetic it is unique, note that 5 is absent, and conclude.

Takeaway. The theorem's name is part of the marking scheme.

Two numbers, two questions

Biggest that fits, smallest they fit into

HCF: the largest number that divides all of them.

Takeaway. HCF looks down at factors; LCM looks up at multiples.

The HCF rule

Shared primes only, lowest level

HCF = product of the smallest power of each prime common to all the numbers.

Takeaway. No shared prime, no place in the HCF.

The LCM rule

Every prime, highest level

LCM = product of the greatest power of every prime that appears in any of the numbers.

Takeaway. Leave out no prime from any number.

Remember which power goes where

Humble and greedy

HCF is humble: smallest powers, shared primes only.

Takeaway. Mixing these two up is the most common error in this topic.

The product rule

Works for pairs

$12 \times 840 = 10080 = 84 \times 120$ ✓

Takeaway. Know any three of HCF, LCM, a and b, and you can find the fourth.

Three numbers break the product rule

A classic assertion–reason trap

HCF(12, 18, 30) × LCM = 6 × 180 = 1080, but 12 × 18 × 30 = 6480.

Takeaway. The middle power of each prime gets lost.

LCM from a given HCF

Divide first, multiply later

Takeaway. Cancel before you multiply.

HCF and LCM of three numbers

The same rules, one more number

Takeaway. 12, 18, 30 give HCF 6 and LCM 180.

HCF story or LCM story?

Read the story, pick the tool

Takeaway. Split → HCF. Meet again → LCM.

Tiling a floor

420 by 130 cm, no cutting

Takeaway. Largest size that fits exactly is always the HCF.

When bells ring together

40, 60 and 90 minutes

Takeaway. Coinciding again is always the LCM.

The word that fools students

'Maximum' does not mean LCM

"The maximum number of students in each row" asks for the largest number dividing each class size — the HCF.

Takeaway. Ask: am I splitting or am I waiting? Not: is the word big or small?

HCF with remainders

Subtract the remainder first

The largest number dividing 245 and 1029 with remainder 5 each time is 16, since HCF(240, 1024) = 16.

Takeaway. The number divides a − r and b − r exactly.

The HCF always divides the LCM

A quick filter for MCQs

HCF 18 and LCM 380 cannot belong to two numbers: 18 does not divide 380.

Takeaway. Check divisibility before you calculate anything.

Co-prime numbers

HCF of 1

Consecutive numbers such as 20 and 21 are always co-prime.

Takeaway. No shared prime means LCM = a × b.

The easy case

When one divides the other

If a divides b, HCF(a, b) = a and LCM(a, b) = b. HCF(15, 60) = 15, LCM = 60.

Takeaway. Spot it and skip the factorising.

Runners meeting at the start

Sonia and Ravi on the track

Laps take 18 and 12 minutes. They meet at the start after LCM(18, 12) = 36 minutes.

Takeaway. NCERT's own track problem is an LCM question.

Show your working

The factorisations carry marks

In 2- and 3-mark questions, writing both prime factorisations usually earns a mark on its own. Then state HCF and LCM separately.

Takeaway. Skip the factorisations and you skip a mark.

Finding x in an LCM

Unknown powers

LCM of $2^x \times 3^2$ and $2^3 \times 3^4 \times 5$ is $2^5 \times 3^4 \times 5$. The LCM takes the larger power of 2, so $x = 5$.

Takeaway. Compare powers prime by prime.

Some numbers are not fractions

The diagonal no ruler can hit

A 1 m square tile has diagonal √2 m. $\left(\frac{99}{70}\right)^2 \approx 2.0002$ — close, but no fraction squares to exactly 2.

Takeaway. √2 is irrational, and it can be proved.

Rational and irrational

p over q, q not zero

Takeaway. Together they make the real numbers.

Some roots are rational

A root sign is not enough

$\sqrt{16} = 4$, $\sqrt{\frac{4}{9}} = \frac{2}{3}$ and $\sqrt{0.09} = 0.3$ are all rational.

Takeaway. √n is irrational only when n is not a perfect square.

If p divides a², p divides a

The engine of every proof

For a prime p: if p divides $a^2$, then p divides a.

Takeaway. Quote this theorem by name in your proof.

p must be prime

It fails for 4

4 divides $36 = 6^2$, but 4 does not divide 6.

Takeaway. Check that p is prime before using it.

Proof by contradiction

Assume the opposite, then break it

Takeaway. The contradiction proves the assumption wrong.

√2 is irrational

The proof examiners expect

Takeaway. Every step carries part of the marks.

Assume, Square, Share, Contradict

Four words for the proof

Assume √p = a/b in lowest terms. Square. Show p divides a, then b, so they share p. Contradict 'lowest terms'.

Takeaway. Works for √2, √3, √5 and any √prime.

Don't forget 'co-prime'

The word that makes the proof work

The contradiction is that a and b share a factor. If you never said they were co-prime, there is nothing to contradict.

Takeaway. Write 'a and b are co-prime' in the first line.

rational ± irrational

Get the root on its own

$5 - \sqrt{3} = \frac{a}{b} \Rightarrow \sqrt{3} = \frac{5b - a}{b}$, which would be rational.

Takeaway. Isolate the root, then point at the contradiction.

Proving 3 + 2√5 irrational

A board favourite

Takeaway. Subtract 3b/b, not just 3, and keep the b.

non-zero rational × irrational

But not by zero

$7\sqrt{5} = \frac{a}{b}$ gives $\sqrt{5} = \frac{a}{7b}$: impossible. So $7\sqrt{5}$ is irrational.

Takeaway. Multiply or divide an irrational by a non-zero rational and it stays irrational.

Irrational with irrational

No rule for two irrationals

Takeaway. Work each one out — the answer can go either way.

Sorting numbers in an MCQ

Simplify before you judge

$\frac{2\sqrt{3}}{\sqrt{12}} = 1$ and $(\sqrt{5} - 2)(\sqrt{5} + 2) = 1$ look irrational but are not.

Takeaway. Always simplify first, then decide.

Why 1/√2 is irrational

Flip it and the root appears

Assume $\frac{1}{\sqrt{2}} = \frac{a}{b}$. Then $\sqrt{2} = \frac{b}{a}$, which would be rational. Contradiction.

Takeaway. A reciprocal of an irrational is irrational too.

√2 on your desk

Why A4 halves into A5

A-series paper has sides in the ratio 1 : √2. Fold a sheet in half and the new sheet has the same shape.

Takeaway. An irrational ratio is behind every A4 sheet.

The Pythagoreans and √2

A 2500-year-old shock

The Pythagoreans believed every length was a ratio of whole numbers. The irrationality of √2 broke that belief.

Takeaway. The proof you learn is one of the oldest in mathematics.

How the 3-mark proof is marked

Name each step's reason

Takeaway. Each link of the chain earns part of the 3 marks.