Multiple choice

If x is a number of five-digits which when divided by 8, 12, 15 and 20 leaves respectively 5, 9, 12 and 17 as remainders, then find x such that it is the lowest such number.

  1. 10017

  2. 10057

  3. 10097

  4. 10077
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D Correct answer
Explanation

The divisors are 8, 12, 15, 20. The remainders are 5, 9, 12, 17. Note that (divisor - remainder) is constant at 3. LCM(8, 12, 15, 20) = 120. The number is 120k - 3. We need a 5-digit number, so 120k - 3 >= 10000. 120k >= 10003, k >= 83.35. For k=84, 120(84) - 3 = 10080 - 3 = 10077.

AI explanation

Notice that the difference between each divisor and its respective remainder is 3. If you add 3 to the number, it becomes perfectly divisible by 8, 12, 15, and 20. The least common multiple of 8, 12, 15, and 20 is 120, making the lowest such base number 123. Since 123 is a three-digit number, add 120 repeatedly until you reach the lowest five-digit number, which is 10077.

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