Multiple choice

What is the sum of all two-digit numbers which when divided by $3$ leave $2$ as the remainder?

  1. $1565$
  2. $1585$
  3. $1635$
  4. $1655$
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C Correct answer
Explanation

Two-digit numbers leaving remainder 2 when divided by 3 are of the form 3n + 2. The sequence is 11, 14, ..., 98. This is an arithmetic progression with a = 11, l = 98, d = 3. Number of terms n: 98 = 11 + (n-1)3 => 87 = (n-1)3 => 29 = n-1 => n = 30. Sum = (n/2)(a + l) = (30/2)(11 + 98) = 15 * 109 = 1635.

AI explanation

The two-digit numbers leaving a remainder of 2 when divided by 3 form an arithmetic progression starting at 11 and ending at 98 with a common difference of 3. The number of terms is (98 minus 11) divided by 3, plus 1, which equals 30. Using the arithmetic series sum formula, the sum equals 30 divided by 2, multiplied by the sum of the first and last terms (11 plus 98). This calculates to 15 multiplied by 109, giving a total sum of 1635.

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