Multiple choice

The number of positive integers, which divide 2006 to leave a remainder of 26 is _________.

  1. 16

  2. 20

  3. 22

  4. 34
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C Correct answer
AI explanation

If a divisor leaves a remainder of 26, it must be greater than 26, and the number being divided equals the divisor multiplied by some quotient plus 26. This means the required divisor must perfectly divide 2006 minus 26, which is 1980. We find the number of factors of 1980 that are strictly greater than 26 by first finding its prime factorization, which is 2 squared times 3 squared times 5 times 11, yielding 36 total factors. Since 1980 divided by 26 is approximately 76, the factors greater than 26 are exactly the factors of 1980 that are less than 76, which count to 22.

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