Multiple choice

Find the sum of all two-digit numbers that give a remainder of $3$ when they are divided by $7$.

  1. $686$
  2. $676$
  3. $666$
  4. $656$
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B Correct answer
Explanation

The numbers are 10, 17, ..., 94. This is an arithmetic progression with a = 10, d = 7, and last term l = 94. 94 = 10 + (n-1)7 => 84 = (n-1)7 => n-1 = 12 => n = 13. Sum = (n/2)(a + l) = (13/2)(10 + 94) = (13/2)(104) = 13 * 52 = 676.

AI explanation

The smallest two-digit number leaving a remainder of 3 when divided by 7 is 10, and the largest is 94. These numbers form an arithmetic progression with a common difference of 7, and the total number of terms is ((94 minus 10) divided by 7) plus 1, which equals 13. Using the arithmetic series sum formula, the sum equals the number of terms divided by 2, multiplied by the first term plus the last term. This calculates to (13 divided by 2) multiplied by (10 plus 94), resulting in 676.

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