A and B each can do a work in $75$ days. They began the work together but A left after some time and B completed the remaining work in $23$ days. After how many days of the start of the work did A leave?

Let the total work be $75$ units (since each takes $75$ days individually).
* Efficiency of A = $\frac{75}{75}=1\text{ unit/day}$
* Efficiency of B = $\frac{75}{75}=1\text{ unit/day}$

Let $x$ be the number of days A and B worked together before A left.
During these $x$ days, both worked together, so the work done is:

$$\text{Work done by (A + B)}=(1+1)\times x=2x\text{ units}$$

After A left, B completed the remaining work alone in $23$ days:

$$\text{Work done by B alone}=1\times 23=23\text{ units}$$

The total work is equal to $75$ units:

$$2x+23=75$$

$$2x=75-23$$

$$2x=52$$

$$x=26\text{ days}$$

Therefore, A left after $26$ days from the start of the work.