To achieve any possible positive integer, one must be able to achieve £1 using the only the deposit amount (£x) and the withdrawal amount (£y). This allows the amount in the bank account to be able to changed by any whole positive number, for all such numbers are multiples of £1. This can be expressed in the following equation:
ax-by=1, where a and b stand for the deposit amount and the withdrawal amount respectively.
This therefore, restricts the values of x and y based on the following rules:
- x cannot be equal to y unless they both equal 1, for when any multiple of either are subtracted from the other it results in a multiple of their value or 0:
When x=y, ax+by=cx or cy or 0. In the case of x and y =£1, cx or cy could be equivalent to 1.
- x and y cannot both be even numbers and/or 0. This is because when a multiple of an even number is subtracted by a multiple of another even number, the result is always an even number meaning that the result cannot be £1:
When x is even AND y is even, ax-by must also be even.
These are the only 2 rules restricting x and y when one hopes to change their "Strange Bank Account" by any value. x and y can be any whole positive value as long as they both follow the rules mentioned above.