Generalised Forms of Numbers and Digit Reversal Invariants
1. Generalised Place Value Representation
- 2-digit number $ab$: $10a + b$ (where $a \in \{1,\dots,9\}$ and $b \in \{0,\dots,9\}$).
- 3-digit number $abc$: $100a + 10b + c$.
- 4-digit number $abcd$: $1000a + 100b + 10c + d$.
DIGIT REVERSAL INVARIANTS
Let number be N = 10a + b and Reversal be R = 10b + a:
1. SUM: N + R = (10a + b) + (10b + a) = 11(a + b) ===> ALWAYS DIVISIBLE BY 11!
2. DIFFERENCE: N - R = (10a + b) - (10b + a) = 9(a - b) ===> ALWAYS DIVISIBLE BY 9!
2. Three-Digit Reversal Invariant
Let $N = 100a + 10b + c$ and reversed number $R = 100c + 10b + a$ (assume $a > c$):
$$N - R = (100a + 10b + c) - (100c + 10b + a) = 99a - 99c = \mathbf{99(a - c)}$$
- The difference is always divisible by 99 (and therefore by 9, 11, and 33)!
- The quotient when divided by 99 is $(a - c)$, the difference between the first and last digits.
