What is the sum of the digits of the least number which, when divided by 15,…

2019

What is the sum of the digits of the least number which, when divided by 15, 54, 16, and 12, leaves the same remainder 4 in each case and is divisible by 19?

  1. A.

    13

  2. B.

    14

  3. C.

    15

  4. D.

    16

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Correct answer: A

Let the required number be N. Since it leaves remainder 4 when divided by 15, 54, 16, and 12, N - 4 must be a multiple of their LCM.

  1. LCM: LCM(15, 54, 16, 12) = 2^4 x 3^3 x 5 = 2160

  2. Number form: N = 4 + 2160k

  3. Divisibility by 19: 4 + 2160k ≡ 0 (mod 19). Since 2160 ≡ 13 (mod 19), we get 13k ≡ 15 (mod 19)

  4. Solve for k: the inverse of 13 modulo 19 is 3, so k ≡ 45 ≡ 7 (mod 19)

  5. Smallest number: N = 4 + 2160 x 7 = 15124

  6. Digit sum: 1 + 5 + 1 + 2 + 4 = 13

Answer: 13

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