Multiple choice

Determine the smallest 5-digit number that, when divided by 72, 84, and 48 respectively, leaves remainders of 53, 65, and 29.

  1. 10,016

  2. 10,061

  3. 10,601

  4. 10,610

  5. 16,001
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B Correct answer
Explanation

The differences between divisors and remainders are: 72-53=19, 84-65=19, 48-29=19. Find the LCM of 72, 84, 48, which is 1008. Multiples of 1008 are 10080, 11088, etc. The smallest 5-digit number is 10080. Subtracting the common difference 19 gives 10080 - 19 = 10061.

AI explanation

First, subtract the respective remainders from the divisors to find the core divisors, giving 72 - 53 = 19, 84 - 65 = 19, and 48 - 29 = 19, which means the number can be expressed as a multiple of the least common multiple (LCM) of 72, 84 and 48 minus 19. The LCM of 72, 84, and 48 is 1008. Dividing the smallest 5-digit number, 10000, by 1008 gives a quotient of 9, so the next multiple is 10 * 1008 = 10080, and subtracting 19 gives 10061. The correct answer is 10061.

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