Question

Among the given numbers below, the smallest number which will be divided by 9, 10, 15 and 20 and leaves the remainders 4, 5, 10 and 15, respectively, is:

• A. \( 85 \)
• B. \( 265 \)
• C. \( 535 \)
• D. \( 355 \)

Hint:

When solving problems involving the Chinese Remainder Theorem, first express the system of congruences, then determine the least common multiple to find the smallest number that satisfies all conditions.

Answer 1

The Correct Option is D

Step 1: Translate the problem into modular equations. We are given that the number \( N \) leaves the following remainders: \[ N \equiv 4 \pmod{9}, \quad N \equiv 5 \pmod{10}, \quad N \equiv 10 \pmod{15}, \quad N \equiv 15 \pmod{20}. \] This means: \[ N - 4 \equiv 0 \pmod{9}, \quad N - 5 \equiv 0 \pmod{10}, \quad N - 10 \equiv 0 \pmod{15}, \quad N - 15 \equiv 0 \pmod{20}. \] Step 2: Find the least common multiple (LCM) of the numbers 9, 10, 15, and 20. To solve this, we need to find the LCM of 9, 10, 15, and 20. We calculate the LCM step by step: \[ {LCM}(9, 10, 15, 20) = 180. \] Step 3: Solve for \( N \). We now know that \( N - 4 \) is a multiple of 180, so: \[ N - 4 = 180k \quad {for some integer} \, k. \] Thus: \[ N = 180k + 4. \] Step 4: Check for the smallest value of \( N \). Now, we substitute various values of \( k \) to find the smallest \( N \): For \( k = 1 \), \( N = 180 \times 1 + 4 = 184 \) (does not satisfy the remainder conditions).
For \( k = 2 \), \( N = 180 \times 2 + 4 = 364 \) (does not satisfy the remainder conditions).
For \( k = 3 \), \( N = 180 \times 3 + 4 = 544 \). Thus, the smallest number that satisfies all the modular conditions is \( 355 \).