Question : What is the sum of the digits of the least number which when divided by 15,18, and 36 leaves the same remainder 9 in each case and is divisible by 11?
Option 1: 15
Option 2: 16
Option 3: 18
Option 4: 17
Correct Answer: 18
Solution :
The least number which leaves the same remainder when divided by 15, 18, and 36 is the least common multiple (LCM) of these numbers plus the remainder.
15 = 3 × 5
18 = 2 × 3
2
36 = 2
2
× 3
2
The LCM of 15, 18, and 36 = 2
2
× 3
2
×5 = 180
When divided by these numbers, the least number that leaves a remainder of 9 is 180k + 9, where k is some positive integer.
However, this number must also be divisible by 11. The smallest number of 180k + 9 divisible by 11 is 1089.
The sum of the digits of 1089 is 1 + 9 + 8 = 18.
Hence, the correct answer is 18.
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