A certain number when divided by 222 leaves $a$ remainder 35, another number when divided by 407 leaves $a$ remainder 47. What is the remainder when the sum of these two numbers is divided by 37?

Step 1 (Formulate Congruences): Let the first number be $N_1$ and the second be $N_2$. We are given:

• $N_1\equiv 35(\mathrm{mod}222)$

• $N_2\equiv 47(\mathrm{mod}407)$

We want to find the remainder of their sum $N_1+N_2$ modulo 37.

Step 2 (Reduce Moduli to 37): Notice the relationship between the moduli and 37:

• $222=37\times 6⟹222\equiv 0(\mathrm{mod}37)$

• $407=37\times 11⟹407\equiv 0(\mathrm{mod}37)$

Therefore, we can reduce the congruences directly modulo 37:

1. $N_1\equiv 35(\mathrm{mod}37)$

2. $N_2\equiv 47\equiv 47-37=10(\mathrm{mod}37)$

Step 3 (Evaluate the Sum): Add the two congruences:

Reduce 45 modulo 37:

Step 4 (Conclusion): The remainder of the sum is exactly 8.