Step 1: Represent the problem as a system of modular equations: n(mod3) = 2 (dividing by 3 gives a remainder of 2) n(mod5) = 4 (dividing by 5 gives a remainder of 4) Step 2: Find the lowest number where each of the original divisors (3 and 5) give a remainder 1 when divided by the other divisor: 5(mod3) = 1, we want 10. (10 is 3x+1 when x = 3) 3(mod5) = 1, we want 6. (6 is 5x+1 when x = 1) Step 3: Multiply the original remainders of 2 and 4 by the corresponding lowest common multiples found in the previous section and add them together: (2x10) + (6x4) = 44 Step 4: Multiply the original divisors (3x5 = 15) and add to the previous answer until it falls within the desired range. 44 + 15 = 59, our answer.