Multiple choice

What is the positive difference between the smallest and largest 5-digit numbers which, when divided by 2, 3, 5, and 7 respectively, leave the same remainder for all four divisors?

  1. 88980

  2. 89760

  3. 89880

  4. 88990

  5. a
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C Correct answer
Explanation

LCM(2,3,5,7) = 210. Numbers are of form 210k + r. Smallest 5-digit number is 10000. 10000/210 = 47.6, so 210*48 = 10080. Largest is 99999/210 = 476.19, so 210*476 = 99960. Difference = 99960 - 10080 = 89880.

AI explanation

If a number leaves the same remainder when divided by 2, 3, 5 and 7, the difference between any two such numbers must be a multiple of their least common multiple (LCM). The LCM of 2, 3, 5 and 7 is 210, meaning the positive difference between the smallest and largest 5-digit numbers with this property is the greatest 5-digit multiple of 210. To find this, divide the largest 5-digit number, 99999, by 210 to get a quotient of 476 with a remainder of 39; multiplying 476 by 210 gives 99960, but the difference between the largest (99960) and smallest (10080) 5-digit multiples is 89880. The answer is 89880.

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