Divisibility Shortcuts

• Fast rules to check if a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10, and 11 — without doing long division.
• How to use digit sums and alternating digit sums to crack missing-digit puzzles.
• How competition problems combine two or more rules at once.

1. Last-digit rules — divisible by 2 if the last digit is even; by 5 if it ends in 0 or 5; by 10 if it ends in 0.
2. Last-two/three-digit rules — divisible by 4 if the last 2 digits form a number divisible by 4; by 8 if the last 3 digits are divisible by 8.
3. Digit-sum rules — divisible by 3 if the digit sum is divisible by 3; by 9 if the digit sum is divisible by 9.
4. Alternating-sum rule for 11 — take digits from the right, alternately add and subtract; if the result is divisible by 11 (including 0), so is the number.
5. Combined rule — divisible by 6 means divisible by BOTH 2 and 3.

Find the missing digit in 4_28 so that the number is divisible by 9.

Step 1 — add the known digits: 4 + 2 + 8 = 14
Step 2 — the total digit sum (14 + missing digit) must be a multiple of 9
Step 3 — test missing digits 0-9: 14 + 4 = 18, which is divisible by 9
Step 4 — the missing digit is 4, giving the number 4428

• Applying the digit-sum rule (for 3 or 9) to divisibility by 2, 4, 5, 8, or 10 — those depend on the last digits, not the full digit sum.
• Forgetting that a number can satisfy divisibility by 3 with MORE THAN ONE possible missing digit — always check all values 0-9 in olympiad problems.
• Mixing up the alternating sum direction for the rule of 11 (it doesn't matter which end you start from, as long as you alternate consistently).

• Which digit makes 7_35 divisible by 3? (Check all valid answers — there may be more than one!)
• Is 9,834 divisible by 11? Use the alternating sum rule.
• A 3-digit number is divisible by both 4 and 9. What must be true about its last 2 digits and its digit sum?

• 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.