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2024 AMC 10B Problem 8

Problem 8 of 25EasierNumber TheoryCounting & Probability

Let NN be the product of all the positive integer divisors of 42.42. What is the units digit of N?N?

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Solution

Since 42=23742 = 2 \cdot 3 \cdot 7 has (1+1)3=8(1+1)^3 = 8 divisors, we can pair each divisor with its complement, so N=4282=424.N = 42^{\frac{8}{2}} = 42^4. Only the units digit matters, and 24=162^4 = 16 ends in 6,6, so 42442^4 does too. Therefore, the answer is D.

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Concepts: factor counting · units digit · pairing and grouping

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.