Multiple choice

What will be the remainder if the number $(7)^{2017}$ is divided by $25$

  1. $1$
  2. $7$
  3. $18$
  4. $24$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

7^2 = 49, which is 50 - 1. So 7^2017 = 7 * (7^2)^1008 = 7 * (49)^1008 = 7 * (50 - 1)^1008. Using binomial expansion, (50 - 1)^1008 = 50k + 1. Thus, 7 * (50k + 1) = 350k + 7. The remainder when divided by 25 is 7.

AI explanation

To find the remainder of 7 raised to the power of 2017 divided by 25, we use cyclicity and exponent rules. First, find the remainder of the base 7 divided by 25, which is 7. Notice that 7 squared is 49, and 49 divided by 25 leaves a remainder of negative 1. Applying the power law of remainders, we evaluate 7 raised to the power of 2016 multiplied by 7. Since negative 1 raised to the even power 2016 equals 1, the remainder becomes 1 multiplied by 7, giving a final result of 7.

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