Multiple choice

If the fractional part of the number $\dfrac {2^{403}}{15}$ is $\dfrac {k}{15}$, then $k$ is equal to

  1. $14$
  2. $6$
  3. $4$
  4. $8$
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D Correct answer
Explanation

We need 2^403 mod 15. Note 2^4 = 16 = 1 mod 15. So 2^403 = 2^(4*100 + 3) = (2^4)^100 * 2^3 = 1^100 * 8 = 8 mod 15. Thus k = 8.

AI explanation

To find the remainder of 2^403 divided by 15, we use the pattern of remainders for powers of 2 modulo 15. The sequence of remainders is 2, 4, 8, and 1, which then repeats every four powers. Dividing the exponent 403 by 4 gives a remainder of 3, so 2^403 has the same remainder as 2^3, which is 8. The fractional part of the number is the remainder divided by the denominator, giving 8/15, therefore k equals 8.

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