2024 AMC 10B Problems/Problem 8: Difference between revisions

Latest revision as of 21:20, 6 October 2025

Problem

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

$\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$

Solution 1

The factors of $42$ are $1, 2, 3, 6, 7, 14, 21, 42$ . Multiply the unit digits to get $\boxed{\textbf{(D) } 6}$

Solution 2

The product of the factors of a number $n$ is $n^\frac{\tau(n)}{2}$ , where $\tau(n)$ is the number of positive divisors of $n$ . We see that $42 = 2^1 \cdot 3^1 \cdot 7^1$ has $(1+1)(1+1)(1+1) = 8$ factors, so the product of the divisors of $42$ is

$42^{\frac{8}{2}} \equiv 42^4 \equiv 2^4 \equiv 16 \equiv \boxed{\textbf{(D) } 6} \pmod{10}.$

Video Solution by 1 Scholars Foundation (Easy to Understand)

https://youtu.be/T_QESWAKUUk?si=E8c2gKO-ZVPZ2tek&t=201

Video Solution 2 by Pi Academy (Fast and Easy)

https://youtu.be/QLziG_2e7CY

Video Solution 3 by SpreadTheMathLove

https://www.youtube.com/watch?v=24EZaeAThuE

Video Solution by TheBeautyofMath

https://youtu.be/ZaHv4UkXcbs?t=603

~IceMatrix

2024 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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-If Bob billy and bilal all eat 2, 3, and 912 watermelons respectively, whats the best way they can eat all of them such that only half of the watermelons have integer divisors less than 2?
+== Problem ==
+Let <math>N</math> be the product of all the positive integer divisors of <math>42</math>. What is the units digit
+of <math>N</math>?
+<math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math>
+== Solution 1 ==
+The factors of <math>42</math> are <math>1, 2, 3, 6, 7, 14, 21, 42</math>. Multiply the unit digits to get <math>\boxed{\textbf{(D) } 6}</math>
+==Solution 2==
+The product of the factors of a number <math>n</math> is <math>n^\frac{\tau(n)}{2}</math>, where <math>\tau(n)</math> is the number of positive divisors of <math>n</math>. We see that <math>42 = 2^1 \cdot 3^1 \cdot 7^1</math> has <math>(1+1)(1+1)(1+1) = 8</math> factors, so the product of the divisors of <math>42</math> is
+<cmath>42^{\frac{8}{2}} \equiv 42^4 \equiv 2^4 \equiv 16 \equiv \boxed{\textbf{(D) } 6} \pmod{10}.</cmath>
+== Video Solution by 1 Scholars Foundation (Easy to Understand)==
+https://youtu.be/T_QESWAKUUk?si=E8c2gKO-ZVPZ2tek&t=201
+== Video Solution 2 by Pi Academy (Fast and Easy) ==
+https://youtu.be/QLziG_2e7CY
+==Video Solution 3 by SpreadTheMathLove==
+https://www.youtube.com/watch?v=24EZaeAThuE
+==Video Solution by TheBeautyofMath==
+https://youtu.be/ZaHv4UkXcbs?t=603
+~IceMatrix
+== See Also ==
+{{AMC10 box|year=2024|ab=B|num-b=7|num-a=9}}
+{{MAA Notice}}

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