Multiple choice

Find the greatest 5-digit number, which when divided by 8, 12, 15 and 20 leaves remainders 5, 9, 12 and 17, respectively.

  1. 99,957

  2. 99,907

  3. 99,757

  4. 90,113

  5. 90,007
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A Correct answer
Explanation

The divisors are 8, 12, 15, 20. LCM(8, 12, 15, 20) = 120. The differences between divisors and remainders are constant: 8-5=3, 12-9=3, 15-12=3, 20-17=3. We need the largest 5-digit number N such that N = 120k - 3. 99999 / 120 = 833 with remainder 39. 120 * 833 = 99960. 99960 - 3 = 99957.

AI explanation

Observing the divisors (8, 12, 15, 20) and their respective remainders (5, 9, 12, 17), we see each divisor minus its remainder equals 3. Therefore, the required number is 3 less than a multiple of the least common multiple of 8, 12, 15 and 20. The LCM of these divisors is 120, so we find the greatest 5-digit multiple of 120, which is 99960. Subtracting 3 from 99960 gives the final result of 99957.

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