Class 6 | Maths
Chapter 5 | Prime Time
Aligned with CBSE | NCERT | School & Olympiad Standards
Focus on What Matters Most
The67Sprint
Factors & Multiples
- Factor: divides a number exactly, no remainder — finite set.
- Multiple: product of a number and a whole number — infinite jumps on a number line.
Prime & Composite Numbers
- Prime: exactly 2 factors — 1 and itself (2, 3, 5, 7...).
- Composite: more than 2 factors (4, 6, 8, 9...). 1 is neither.
Sieve of Eratosthenes
- An ancient grid method: cross out all multiples of 2, then 3, then 5, and so on.
- Whatever remains uncrossed up to your limit is prime.
Perfect Numbers
- A number where the sum of all its factors equals exactly twice the number.
- 28: factors 1,2,4,7,14,28 → sum = 56 = 28×2.
Co-prime & Twin Primes
- Co-prime: share no common factor except 1 — needn't be prime themselves (4 and 9).
- Twin Primes: two primes with a difference of 2 (3 & 5, 11 & 13).
Divisibility Tests
- 2, 5, 10: check the last digit. 4: last 2 digits. 8: last 3 digits.
- 3, 9: sum the digits — divisible if the sum is. 6: must pass both 2 and 3.
Unique Prime Factorisation
- Every composite number breaks down into one unique set of primes, regardless of order.
- 198 = 2 × 3 × 3 × 11, using a Factor Tree or Division Method — always circle down to primes only.
The Golden Formula
For exactly two numbers: HCF × LCM = Product of the numbers. This does not extend to three or more numbers.
The67Sprint
Quick Recall — Sprint Facts
Smallest Prime
2 — the only even prime
1 and 0
Neither prime nor composite
Golden Formula
Only for exactly two numbers
Leap Years
Multiples of 4, except centuries not ÷400
🔑 Key Terms
Factor
A divisor that leaves zero remainder.
Prime Factorisation
Breaking a number into a product of only primes.
Twin Primes
A pair of primes with a difference of 2.
Sieve of Eratosthenes
A grid method that isolates primes by crossing out multiples.
The67Sprint
🏆 Olympiad & Challenger
The Fundamental Theorem of Arithmetic
The formal name for unique prime factorisation — every composite number has exactly one prime breakdown.
Advanced Divisibility (7, 11, 13, 19, 23)
For 11: alternating digit sums differ by a multiple of 11. For 19: double the last digit and add to the rest.
The Pigeonhole Principle
If m+1 items fill n boxes, one box holds at least ⌊m/n⌋+1 items — a core Olympiad reasoning tool.
Invariants
A property that never changes through repeated transformations — like whether a sum stays even, no matter how numbers are combined.
The67Sprint
Odd ≠ Prime
9, 15, and 21 are odd but composite. Always run a divisibility check before declaring a number prime.
Incomplete Factor Trees
Stopping at 6×10 for 60 loses marks — branch every composite endpoint down until only primes remain.
Formula Misuse for 3+ Numbers
HCF × LCM = Product only works for exactly two numbers — never apply it to three or more.
Formatting Prime Factors
List primes from least to greatest for full marks — 198 = 2 × 3 × 3 × 11, not 11 × 2 × 3 × 3.
Jump Jackpot & Idli-Vada
Jump sizes that land exactly on 24 are its factors. In Idli-Vada, "Idli-Vada" lands on common multiples of 3 and 5 — 15, 30, 45.
Packing & Arranging
Packing 12 figs into rows and columns uses every factor pair of 12 — 1×12, 2×6, 3×4.
Traffic Light Syncing
Lights changing every 48s, 72s, and 108s sync again at their LCM — 432 seconds later.
Digital Security
Massive prime numbers are the backbone of RSA encryption that keeps passwords and messages locked.
Co-prime ≠ Both Prime
4 and 9 are both composite, yet co-prime — their only common factor is 1. Co-prime never requires primality.
HCF via Keywords
Splitting into smaller equal groups needs HCF; syncing repeating events needs LCM — read the action, not just the keyword.
The Difference Method Edge Case
If the difference doesn't divide both numbers, check its factors instead — HCF of 15 & 25: difference 10 doesn't divide 15, but its factor 5 does.
Logic Chain: Prime Factorisation
Step 1
Take any composite number and find one factor pair.
Step 2
Check each branch — if prime, circle it; if composite, factor it again.
Step 3
Repeat until every branch ends in a circled prime number.
Step 4
Write the original number as the product of all circled primes, smallest to largest.
🦉
Mentor Byte
"Every composite number carries its own DNA — a unique set of primes hidden inside it. Before reaching for long division, run the divisibility tests first. Speed in this chapter comes from recognising the DNA at a glance, not from calculating it the long way every time."
Parent Tip
Pick a number together and ask your child to guess its prime factors before checking. Guessing first builds the instinct faster than working it out cold.