REVISION SUMMARY: Number Play (Chapter 5)
1. Chapter at a Glance
- Any expression formed by placing + and – signs between four numbers always has the same parity (all even or all odd).
- Even numbers are of two types: multiples of 4 (remainder 0 mod 4) and those with remainder 2 mod 4; their sums follow clear patterns.
- If a divides M and a divides N, then a divides both M + N and M – N.
- If a number is divisible by k, then all its multiples are also divisible by k, and the number is divisible by every factor of k.
- A number is divisible by 9 (or 3) if and only if the sum of its digits is divisible by 9 (or 3); the same holds for repeated digit sums (digital root).
- A number is divisible by 11 if the difference between the sum of digits in odd places and even places (alternating) is a multiple of 11 (including 0).
- Numbers leaving a fixed remainder r when divided by d are of the form dk + r.
- Statements about factors and multiples are classified as Always True, Sometimes True or Never True; general rules involve LCM when two divisors are involved.
2. Definitions, Theorems and Results
- Parity: Even numbers are multiples of 2; odd numbers leave remainder 1 when divided by 2. Switching a sign in an expression changes its value by an even number (2b), so parity is preserved.
- Result on four numbers: All eight expressions a ± b ± c ± d have identical parity (proved by showing each sign change preserves parity or by inductive extension of a ± b having same parity).
- General divisibility property (Always True): If a | M and a | N, then a | (M + N) and a | (M – N). (Proof required in exam via algebra: M = a m, N = a n.)
- Multiples property (Always True): If A is divisible by k, then every multiple of A is divisible by k. (Proof: (k j) × m = k (j m).)
- Factor property (Always True): If A is divisible by k, then A is divisible by every factor of k.
- LCM property (Always True): If A is divisible by both k and m, then A is divisible by LCM(k, m).
- Remainder form: Numbers leaving remainder r when divided by d are exactly the numbers of form d k + r (k ≥ 0, with suitable restriction to keep non-negative).
- Digital root: Repeated sum of digits yields the remainder when the number is divided by 9 (or 9 itself if remainder 0).
No separate lemmas or axioms; all results are proved using algebra or place-value expansion.
3. Formula Sheet
| Expression / Rule |
Meaning |
Symbols |
| dk + r |
All numbers leaving remainder r on division by d |
d = divisor, k = integer ≥ 0, r = remainder (0 ≤ r < d) |
| Sum of digits ≡ number mod 9 |
Divisibility / remainder by 9 |
— |
| Alternating place sum difference |
Divisibility / remainder by 11 |
Odd places minus even places (or vice versa) |
| 4m + 2q = 2(2m + q) |
Always even |
m, q any integers |
| a ± b has same parity |
Parity of sum or difference identical |
a, b any integers |
4. Solved-Example Patterns
- Pattern A (Remainder form): Identify algebraic expression for numbers leaving given remainders (e.g., remainder 3 when divided by 5). Method: Write “multiple of divisor plus/minus fixed difference” → test options or derive dk + r.
- Pattern B (Always/Sometimes/Never): Classify statements using the four general properties above. Method: Translate to multiples (e.g., 8a, 8b), apply the relevant property, give algebraic justification and one example + one counter-example when “sometimes”.
- Pattern C (Divisibility shortcut application): Check divisibility by 3/9/11 without division. Method: Compute digit sum or alternating sum; reduce repeatedly; state remainder or “divisible”.
- Pattern D (Consecutive / sum properties): Find consecutive numbers or check parity/sum divisibility. Method: Express as n, n+1, … or use even/odd and multiple-of-4 cases.
- Pattern E (Cryptarithm): Solve letter-for-digit puzzles. Method: Use place-value constraints, multiplication/division size limits, uniqueness of digits, and divisibility checks on the resulting numbers.
5. Common Mistakes and Exam Pitfalls
- Forgetting that checking divisibility by 4 and 6 is insufficient for 24 (use 3 and 8 instead).
- Sign error when computing alternating sum for divisibility by 11 (must start from units place as + or follow consistent alternation).
- Assuming “if divisible by 9 then divisible by 18” or similar without checking the second factor.
- Writing dk + r without ensuring r < d or allowing negative k when the problem requires positive numbers.
- Missing that digital root 9 means the number is divisible by 9 (remainder 0).
- In “always/sometimes/never” questions, giving only an example without the algebraic generalisation required by the chapter.
- Overlooking that the sum of an odd and even number is never a multiple of 6 (parity contradiction).