Multiple choice

Find the greatest number of five digits which when divided by $4, 6, 14$ and $20$ leaves respectively $1, 3, 11$ and $17$ as remainders.

  1. $99930$
  2. $99960$
  3. $99997$
  4. $99957$
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D Correct answer
Explanation

The differences between divisors and remainders are constant: 4-1=3, 6-3=3, 14-11=3, 20-17=3. Find the LCM of (4, 6, 14, 20) = 420. The largest 5-digit number is 99999. 99999 / 420 = 238 with remainder 39. The largest multiple is 99999 - 39 = 99960. Subtract the common difference 3 to get 99957.

AI explanation

Each divisor minus its respective remainder yields 3, 3, 3 and 3. The LCM of 4, 6, 14 and 20 is 420. The greatest five-digit number is 99999, which divided by 420 gives a remainder of 39. Subtracting 39 from 99999 gives 99960, and subtracting 3 gives 99957. The number is 99957.

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