Chinese Remainder theorem

Question

Exercise : Find a number q, which divided by p_i will give remainder q_i for: p_1 = 12, p_2 = 7, q_1 = 4, q_2 = 5.

Answer 1

You're looking for a number q that is 4 more than a multiple of 12, and 5 more than a multiple of 7.

12 is larger, so satisfy that condition first: the first number 4 more than a multiple of 12 is 4. Now, repeatedly add 12 to your number until you hit on something that is 5 more than a multiple of 7. Since you're adding 12, this will always preserve your original remainder.

Adding 12 gives 4, 16, 28, 40, and 40=5•7+5. So 40 is the smallest positive number satisfying your conditions.