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2025 AMC 12B Problem 3

Below is the professionally curated solution for Problem 3 of the 2025 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.

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Concepts:complex number

Difficulty rating: 1130

3.

What is the value of i(i−1)(i−2)(i−3),i(i-1)(i-2)(i-3), where i=−1?i = \sqrt{-1}?

6−5i6 - 5i

−10i-10i

10i10i

−10-10

1010

Solution:

i(i−1)=i2−i=−1−i,i(i-1) = i^2 - i = -1 - i, and (i−2)(i−3)=i2−5i+6=5−5i.(i-2)(i-3) = i^2 - 5i + 6 = 5 - 5i. Then (−1−i)(5−5i)=−5+5i−5i+5i2=−5−5=−10.(-1-i)(5-5i) = -5 + 5i - 5i + 5i^2 = -5 - 5 = -10.

Thus, the correct answer is D.

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