Under modular conditions

Of all the integers $x > 100$ , such that $x \equiv -4 \pmod {12}$ , only $x = 92 \equiv 2 \pmod {5} \equiv 1 \pmod {13}$ (see table below). Therefore, $a=92 \equiv \boxed{-2} \pmod {94}$ .

Since (a mod 5) = 2 and (a mod 13) = 1, therefore (a mod 65) = 27. So a can be in this arithmetic sequence: 27, 92, 157, ... Since a is less than 100 the only possibilities are: 27 and 92. Now let's check a mod 12: (a mod 12) = -4. To use positive numbers we can say: (a mod 12) = 8. To check 27: (27 mod 12) = 3 and not 8, this is to be rejected. To check 92: (92 mod 12) = 8. Therefore a = 92. Now we have (n mod 94) = a. Therefore (n mod 94) = 92. So n = 92. Since n has to be between -45 and 51, we subtract 94 from n this gives us: n = 92 - 94 = -2.

a=92 so 92=-2 mod 94 .... n=-2