A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

Let the number be N and divisor be D. We have: (1) N ≡ 24 (mod D), (2) 2N ≡ 11 (mod D). From (2), 2N - 11 is divisible by D. Since N = kD + 24, we have 2(kD + 24) - 11 = 2kD + 48 - 11 = 2kD + 37. For this to be divisible by D, 37 must be divisible by D. Therefore, D = 37. We also need D > 24 (since remainder must be less than divisor), which 37 satisfies.