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.
↳ Primes are the building blocks of all whole numbers.
📚 Chapter Learning Kit
Real Numbers · Class 10 Mathematics
English 3 topics 176 leveled MCQs 64 flashcards 3 games Free
Shared by a Veda teacher · Generated with Veda AI · Sep 2026
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📖 Definition
⚠️ Common Mistake
💡 Key Idea
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➗ Formula
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Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. Here's a taste from this kit; the app has the full deck.
What makes a number prime
Two factors, no more, no fewer
↳ 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.
↳ Primes start at 2.
Two numbers, two questions
Biggest that fits, smallest they fit into
HCF: the largest number that divides all of them.
↳ 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.
↳ 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.
↳ Leave out no prime from any number.
📖 Smart notes
What you'll study, topic by topic
1
Primes and the Fundamental Theorem of Arithmetic
Every whole number bigger than 1 is either a prime or is built by multiplying primes together — and there is only ONE set of primes that builds it. That single fact, the Fundamental Theorem of Arithmetic, lets you factor…
- A PRIME has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, … A COMPOSITE has more than two.
- 1 is neither prime nor composite. 2 is the only even prime.
- FUNDAMENTAL THEOREM OF ARITHMETIC: every composite number can be written as a product of primes, and this factorisation is unique apart from…
~35 min · full explanation, examples & memory tricks in the app
2
HCF and LCM by Prime Factorisation
Once two numbers are written as products of primes, their HCF and LCM can be read off in one line: the HCF takes the SMALLEST power of each COMMON prime, the LCM takes the GREATEST power of EVERY prime. For two numbers,…
- HCF = product of the SMALLEST power of each prime COMMON to all the numbers.
- LCM = product of the GREATEST power of EVERY prime that appears in any of the numbers.
- $84 = 2^2 \times 3 \times 7$, $120 = 2^3 \times 3 \times 5$: HCF $= 2^2 \times 3 = 12$, LCM $= 2^3 \times 3 \times 5 \times 7 = 840$.
~40 min · full explanation, examples & memory tricks in the app
3
Irrational Numbers and Their Proofs
√2 cannot be written as a fraction — and you can PROVE it, by assuming it can and watching the assumption fall apart. This topic gives you that proof for √2, √3 and √5, and then the one-line argument that turns them into…
- RATIONAL: can be written as $\frac{p}{q}$ with p, q integers and $q \neq 0$. IRRATIONAL: cannot.
- THEOREM: if a prime p divides $a^2$ (a a positive integer), then p divides a. It rests on the Fundamental Theorem of Arithmetic.
- PROOF BY CONTRADICTION: assume the opposite, reason correctly, reach something impossible, conclude the assumption was false.
~45 min · full explanation, examples & memory tricks in the app
❓ Leveled MCQ practice
Try the smart MCQs from this kit
176 questions laddered from warm-up to topper-level, each with an explanation. A taste:
Which of these numbers is prime?
Beginner
A 91
B 97
C 87
D 51
Show answer & explanation
97
$91 = 7 \times 13$, $87 = 3 \times 29$ and $51 = 3 \times 17$. No prime up to 9 divides 97, so 97 is prime.
Which statement about the number 1 is true?
Beginner
A It is the smallest prime
B It is the smallest composite
C It is neither prime nor composite
D It is both prime and composite
Show answer & explanation
It is neither prime nor composite
A prime has exactly two factors and a composite more than two. 1 has only one factor, so it is neither.
How many even prime numbers are there?
Beginner
A None
B Exactly two
C Infinitely many
D Exactly one
Show answer & explanation
Exactly one
2 is prime. Every larger even number has 1, 2 and itself as factors, so it is composite.
The prime factorisation of 360 is
Beginner
A $2^2 \times 3^2 \times 10$
B $2^3 \times 45$
C $2^3 \times 3^2 \times 5$
D $2^3 \times 3 \times 15$
Show answer & explanation
$2^3 \times 3^2 \times 5$
$360 = 36 \times 10 = 2^2 \times 3^2 \times 2 \times 5 = 2^3 \times 3^2 \times 5$. The other options contain the composites 10, 45 or 15.
🃏 Flashcards
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64 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
🎮 Learning games
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Word Match
True False
Fill Blank
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