253. Find a number which, when divided by 17, 13, and 10, gives remainders of 15, 11, and 3, respectively.

From the treatise "De triangulis omnimodis libri quinque".
253. The problem is reduced to solving the system:

$$
17 x+15=13 y+11=10 z+3
$$

The smallest values: $x=64 ; y=84 ; z=110$.

The sought number is 1103.

This problem was proposed by Regiomontanus to the astronomer Bianchini. The latter, by trial and error, found two solutions: 1103 and 3313.

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-135.jpg?height=546&width=603&top_left_y=1052&top_left_x=361)

Ch. 57. Then Regiomontanus indicated that 1103 is the smallest value, and subsequent values are obtained by adding $17 \cdot 13 \cdot 10=2210$, so the second value will be $1103+$ $+2210=3313$ and so on.