Divisibility & Digit Puzzles

• The full toolkit of divisibility rules (3, 4, 6, 8, 9, 11) and why each one works algebraically, not just as a trick.
• How to find missing digits in a number so that it satisfies several divisibility conditions at once.
• How to prove general divisibility statements (e.g. "the product of any 3 consecutive integers is divisible by 6") using algebra instead of checking examples.

1. Digit-sum rules (3 and 9) — a number is divisible by 3 (or 9) exactly when the sum of its digits is divisible by 3 (or 9). This works because 10 ≡ 1 (mod 3) and (mod 9), so each digit contributes its own value to the remainder.
2. Alternating-sum rule (11) — a number is divisible by 11 exactly when the alternating sum of its digits (from the right, + − + − …) is divisible by 11. This works because 10 ≡ −1 (mod 11).
3. Last-digits rules (4 and 8) — a number is divisible by 4 when its last two digits form a number divisible by 4; by 8 when its last three digits form a number divisible by 8 — because 100 is a multiple of 4 and 1000 is a multiple of 8.
4. Proving general statements — write consecutive integers as n, n+1, n+2, …; among any k consecutive integers exactly one is divisible by k, so their product carries that factor automatically.

Find the digit d so that the 5-digit number 4d652 is divisible by both 9 and 11.

Step 1 — divisible by 9: digit sum 4+d+6+5+2 = 17+d must be a multiple of 9 → d=1 (sum=18)
Step 2 — check divisible by 11 with d=1: number is 41652
Step 3 — alternating sum from the right: 2-5+6-1+4 = 6, not a multiple of 11 — fails
Step 4 — try the next multiple of 9 for d: d=1 was the only single digit giving 17+d divisible by 9 in range 0-9 (17+d=18→d=1)
Step 5 — since d=1 fails the 11-test, no single digit satisfies BOTH conditions simultaneously
Answer — there is no digit d making 4d652 divisible by both 9 and 11 (a good olympiad point: not every "nice" combined condition has a solution!)

• Applying the alternating-sum rule for 11 starting from the wrong end — always fix one consistent direction (right to left is safest) and stick to it.
• Assuming a divisibility rule "roughly" works for other numbers (e.g. trying a digit-sum test for 7) — digit-sum tests only work cleanly for 3, 9 (and alternating-sum for 11) because of how powers of 10 behave mod those numbers.
• In "prove it always divides" problems, checking only 2–3 numerical examples instead of giving a general algebraic proof.

• Find the smallest digit d such that 72d is divisible by 4 (last two digits "2d" must be a multiple of 4 → d=0 gives 20, which works, so d=0).
• Prove that the sum of any three consecutive integers is always divisible by 3.
• Is 9,384 divisible by 11? (Alternating sum from right: 4−8+3−9 = −10, not a multiple of 11, so no.)

Open the Practice tab for graded questions on Divisibility & Digit Puzzles.

• Digit-slider tool: students drag a missing digit 0-9 into a blank in a number and instantly see which divisibility rules pass or fail.
• Mirror / body / home activity: take a real bill, receipt, or vehicle number plate and check which small numbers (3, 4, 9, 11) it is divisible by.
• Voice or text reflection with AI Mentor: explain why bank account numbers and Aadhaar numbers use a "check digit" based on divisibility ideas.

• "Explain the divisibility rule for 9 to a Class 6 student using a real multi-digit number from a shop bill."
• "What is one common mistake students make when combining two divisibility rules for the same missing digit, and how would you catch yourself making it?"
• Stretch: "How does this connect to barcode/ISBN check digits, error-detection in bank account numbers, or computer data checksums?"

• 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.