Prime Factor Puzzles

• How to break any number down into a product of primes (its prime factorisation) using a factor tree.
• How to use prime factorisation to find HCF, LCM, and the total number of divisors of a number.
• The classic olympiad trick: turning a number into a perfect square by multiplying it by the smallest possible number.

1. Prime factorisation — every whole number greater than 1 can be written as a unique product of primes (e.g. 60 = 2 × 2 × 3 × 5).
2. Counting divisors — if a number's prime factorisation is p^a × q^b × ..., the number of divisors is (a+1)(b+1)....
3. HCF from prime factors — multiply the LOWEST power of each COMMON prime.
4. LCM from prime factors — multiply the HIGHEST power of EVERY prime that appears in either number.
5. HCF × LCM = product of the two numbers — a powerful shortcut for checking answers.
6. Perfect-square trick — a number is a perfect square only if every prime in its factorisation appears an EVEN number of times; multiply by the primes with odd powers to fix this.

What is the smallest number that 18 must be multiplied by to become a perfect square?

Step 1 — factorise: 18 = 2 × 3 × 3 = 2^1 × 3^2
Step 2 — the prime 2 has an odd power (1); the prime 3 already has an even power (2)
Step 3 — multiply by 2 to make its power even: 18 × 2 = 36 = 2^2 × 3^2
Step 4 — check: 36 = 6^2, a perfect square. Answer: 2.

• Forgetting to check EVERY prime factor's power when hunting for perfect squares — one leftover odd power ruins it.
• Using the ordinary numbers (not their prime factorisations) to compute HCF/LCM on large values — this is slow and error-prone; always factorise first.
• Believing "HCF × LCM = product" is only a rule for calculators — it is a genuine shortcut you should use to double-check answers.

• Find the prime factorisation of 84.
• How many divisors does 72 have?
• Two numbers have HCF 6 and product 216. Find their LCM.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.