2024 AMC 10B Problem 8

Below is the professionally curated solution for Problem 8 of the 2024 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 10B solutions, or check the answer key.

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Concepts:factor counting units digit pairing and grouping

Difficulty rating: 1310

Let $N$ be the product of all the positive integer divisors of $42.$ What is the units digit of $N?$

$0$

$2$

$4$

$6$

$8$

Solution:

Since $42 = 2 \cdot 3 \cdot 7$ has $(1+1)^3 = 8$ divisors, we can pair each divisor with its complement, so $N = 42^{8/2} = 42^4.$ Only the units digit matters, and $2^4 = 16$ ends in $6,$ so $42^4$ does too. Therefore, the answer is D.

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