Chapter 5: Number Play
Exploring Parity, Divisibility Rules, Algebraic Generalisation, Digital Roots & Cryptarithms
๐ข 1. Parity Invariance
Even/Odd BehaviorTake any 4 consecutive numbers (such as 3, 4, 5, 6). Placing any combination of + or − across the 8 possibilities always yields an even number!
Algebraic Proof: Switching a sign from +b to −b changes the total by (a + b − c − d) − (a − b − c − d) = 2b. Since 2b is always an even number, the overall parity never changes!
• Odd ± Odd = Even
• Even ± Even = Even
• Odd ± Even = Odd
⚡ 2. Pairs to Make Fours
Remainder Class LogicEven numbers divided by 4 leave either remainder 0 (form 4p) or remainder 2 (form 4p + 2).
- Multiple + Multiple: 4p + 4q = 4(p + q) → Multiple of 4.
- Non-Multiple + Non-Multiple: (4p + 2) + (4q + 2) = 4(p + q + 1) → Multiple of 4 (remainders 2 + 2 = 4)!
- Multiple + Non-Multiple: 4p + (4q + 2) = 4(p + q) + 2 → Not a multiple of 4.
๐ 3. Divisibility Theorems
Universal Laws- If a divides M and N, then a divides (M + N) and (M − N).
- If A is divisible by k, then all multiples of A are divisible by k.
- If A is divisible by k, then A is divisible by all factors of k.
- If A is divisible by coprime numbers k and m, then A is divisible by LCM(k, m).
➗ 4. Divisibility by 9 & 3
Place Value Decomposition
Powers of 10 are always 1 more than a multiple of 9:
10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1.
dcba = d(999 + 1) + c(99 + 1) + b(9 + 1) + a
= 9 × (Integer) + (d + c + b + a)
Rule: A number is divisible by 9 (or 3) if and only if the sum of its digits is divisible by 9 (or 3).
⚖️ 5. Divisibility by 11
Alternating Excess & Deficit
Place values alternate between 1 more (+1) and 1 less (−1) than multiples of 11:
1 = 0 + 1, 10 = 11 − 1, 100 = 99 + 1, 1000 = 1001 − 1.
Rule: (Sum of Odd Place Digits) − (Sum of Even Place Digits). If the difference is 0 or a multiple of 11, the number is divisible by 11!
๐ฑ 6. Digital Roots & Roots Club
Mahฤsiddhฤnta HeritageRepeatedly summing digits until a single digit remains yields the Digital Root (e.g., 489710 → 29 → 11 → 2).
- Codified by Aryabhata II (c. 950 CE) in Mahฤsiddhฤnta for checking calculations.
- Digital root equals the remainder when divided by 9 (if the root is 9, the remainder is 0).
๐ Master Divisibility Shortcut Matrix
Summary of testing rules and algebraic foundations from the Ganita Prakash curriculum:
| Divisor | Testing Rule | Algebraic Basis | Example |
|---|---|---|---|
| 2, 5, 10 | Check only the Units Digit (last digit). | 10b, 100c, 1000d are already multiples of 2, 5, 10. | 639210 (Divisible by 2, 5, 10) |
| 4 & 8 | Last 2 digits divisible by 4; Last 3 digits divisible by 8. | 100 is a multiple of 4; 1000 is a multiple of 8. | 2856 (56 ÷ 4, 856 ÷ 8) |
| 3 & 9 | Sum of all digits is divisible by 3 or 9. | 10n leaves remainder 1 for each place value. | 405 (4 + 0 + 5 = 9) |
| 6 | Divisible by both 2 and 3 simultaneously. | LCM(2, 3) = 6 because 2 and 3 are coprime. | 429714 (Even and sum = 27) |
| 11 | (Sum of Odd Places) − (Sum of Even Places) is 0 or multiple of 11. | 10n alternates as (11k + 1) and (11k − 1). | 90904 (13 − 2 = 11) |
| 36 & 44 | Divisible by (4 and 9) or (4 and 11) respectively. | Must use coprime factor pairs. | 45036 (Divisible by 36) |
๐ญ 7. Digits in Disguise (Cryptarithms)
Letter-digit puzzles where each distinct letter represents a unique digit (0–9) and the first digit is never 0:
• A1 + 1B = B0 → A = 7, B = 9 (71 + 19 = 90).
• ON + ON + ON = PO → O = 3, N = 1, P = 9 (31 × 3 = 93).
• PQ × 8 = RS → 12 × 8 = 96 (Unique non-repeating digits).
• JK × 6 = KKK → 74 × 6 = 444 (J = 7, K = 4).
๐ฎ 8. Navakankari & Remainder Puzzles
Navakankari (Nine Men's Morris / Sฤlu Mane Aศa): Traditional Indian alignment strategy game using 9 pawns to make lines of three and block opponent moves.
"Grouped by 2, 3, or 5 leaves remainder 1; divisible perfectly by 7; total < 100."
LCM(2, 3, 5) = 30 → Numbers of the form 30k + 1.
Candidates: 31, 61, 91. Since 91 = 7 × 13, the answer is 91 pebbles!
⚠️ Common Pitfalls & Conceptual Traps
Non-Coprime Factors Trap
Checking divisibility by composite non-coprime factors fails! For instance, 12 is divisible by 4 and 6, but not by 24. Always check coprime pairs (e.g., 3 and 8 for 24).
Digit Reversal Invariance
Reversing digits preserves divisibility only for 3 and 9 (sum of digits remains identical). It does not hold for 2, 4, 5, 8, 10, or 11 where place position determines the value!
Leading Zero & Root 9
In multi-digit letter representations like abc, the first letter a ≠ 0. Also, a digital root of 9 corresponds to a remainder of 0 when divided by 9.