Multiple choice

The sum of all $2-$digit numbers which leave remainder $1$ when divided by $3$ is

  1. $1616$
  2. $1602$
  3. $1605$

  4. None of these
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C Correct answer
Explanation

2-digit numbers leaving remainder 1 when divided by 3 are of the form 3k + 1. The smallest is 10 (3*3+1) and the largest is 97 (3*32+1). This is an arithmetic progression: 10, 13, ..., 97. Number of terms n = (97-10)/3 + 1 = 29 + 1 = 30. Sum = (n/2) * (first + last) = (30/2) * (10 + 97) = 15 * 107 = 1605.

AI explanation

The smallest 2-digit number leaving a remainder of 1 when divided by 3 is 10, and the largest is 97. These numbers form an arithmetic progression with the first term 10, the last term 97, and a common difference of 3. Using the formula for the number of terms, n equals ((97 minus 10) divided by 3) plus 1, which gives 30 terms. The sum of an arithmetic progression is n times (first term plus last term) divided by 2, so 30 times (10 plus 97) divided by 2 equals 1605.

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