# Anomalous Cancellation/Examples/3544 over 7531

## Theorem

The fraction $\dfrac {3544} {7531}$ exhibits the phenomenon of anomalous cancellation:

$\dfrac {3544} {7531} = \dfrac {344} {731}$

as can be seen by deleting the $5$ from both numerator and denominator.

This is part of a longer pattern:

$\dfrac {344} {731} = \dfrac {3544} {7531} = \dfrac {35544} {75531} = \cdots$

## Proof

 $\displaystyle \frac {355 \cdots 544} {755 \cdots 531}$ $=$ $\displaystyle \paren {3 \times 10^n + \paren {\sum_{i \mathop = 2}^{n - 1} 5 \times 10^i} + 44} \Big / \paren {7 \times 10^n + \paren {\sum_{i \mathop = 2}^{n - 1} 5 \times 10^i} + 31}$ $\displaystyle$ $=$ $\displaystyle \paren {3 \times 10^n + \paren {500 \times \sum_{i \mathop = 0}^{n - 3} \times 10^i} + 44} \Big / \paren {7 \times 10^n + \paren {500 \times \sum_{i \mathop = 0}^{n - 3} \times 10^i} + 31}$ $\displaystyle$ $=$ $\displaystyle \paren {3 \times 10^n + \paren {500 \times \frac {10^{n - 2} - 1} {10 - 1} } + 44} \Big / \paren {7 \times 10^n + \paren {500 \times \frac {10^{n - 2} - 1} {10 - 1} } + 31}$ Sum of Geometric Sequence $\displaystyle$ $=$ $\displaystyle \dfrac {3 \times 10^n \times \paren {10 - 1} + 500 \times \paren {10^{n - 2} - 1} + 44 \times \paren {10 - 1} } {7 \times 10^n \times \paren {10 - 1} + 500 \times \paren {10^{n - 2} - 1} + 31 \times \paren {10 - 1} }$ simplifying $\displaystyle$ $=$ $\displaystyle \dfrac {27 \times 10^n + 500 \times 10^{n - 2} - 104} {63 \times 10^n + 500 \times 10^{n - 2} - 221}$ simplifying $\displaystyle$ $=$ $\displaystyle \dfrac {2700 \times 10^{n - 2} + 500 \times 10^{n - 2} - 104} {6300 \times 10^{n - 2} + 500 \times 10^{n - 2} - 221}$ simplifying $\displaystyle$ $=$ $\displaystyle \dfrac {3200 \times 10^{n - 2} - 104} {6800 \times 10^{n - 2} - 221}$ simplifying $\displaystyle$ $=$ $\displaystyle \dfrac {8 \times \paren {400 \times 10^{n - 2} - 13} } {17 \times \paren {400 \times 10^{n - 2} - 13} }$ factoring $\displaystyle$ $=$ $\displaystyle \dfrac 8 {17}$ factoring $\displaystyle$ $=$ $\displaystyle \dfrac {8 \times 43} {17 \times 43}$ $\displaystyle$ $=$ $\displaystyle \dfrac {344} {731}$

$\blacksquare$