Question

The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15, and 18 is:

• A. 74
• B. 94
• C. 184
• D. 364

Hint:

When solving congruences involving multiple moduli, use the LCM of the moduli to simplify the problem.

Answer 1

The Correct Option is B

Step 1: Formulate the congruence condition.
We need \( 7n - 4 \) to be divisible by 6, 9, 15, and 18. \[ 7n \equiv 4 \, (\text{mod} \, \text{lcm}(6, 9, 15, 18)) \] LCM = 90 \[ 7n \equiv 4 \, (\text{mod} \, 90) \] Solving for \( n \), \( n \equiv 10 \, (\text{mod} \, 90) \)

Step 2: Find the smallest such \( n \).
The smallest \( n \) satisfying this condition is \( n = 10 \). \[ 7 \times 10 = 70 + 24 = 94 \]