Difference between revisions of "2022 AMC 8 Problems/Problem 3"

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(Video Solution)
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~MathFun1000
 
~MathFun1000
  
==Video Solution==
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==Video Solution 1==
 
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=142
 
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=142
  

Revision as of 01:12, 20 January 2023

Problem

When three positive integers $a$, $b$, and $c$ are multiplied together, their product is $100$. Suppose $a < b < c$. In how many ways can the numbers be chosen?

$\textbf{(A) } 0 \qquad \textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

Solution 1

The positive divisors of $100$ are \[1,2,4,5,10,20,25,50,100.\] It is clear that $10\leq c\leq50,$ so we apply casework to $c:$

  • If $c=10,$ then $(a,b,c)=(2,5,10).$
  • If $c=20,$ then $(a,b,c)=(1,5,20).$
  • If $c=25,$ then $(a,b,c)=(1,4,25).$
  • If $c=50,$ then $(a,b,c)=(1,2,50).$

Together, the numbers $a,b,$ and $c$ can be chosen in $\boxed{\textbf{(E) } 4}$ ways.

~MRENTHUSIASM

Solution 2

The positive divisors of $100$ are \[1,2,4,5,10,20,25,50,100.\] We can do casework on $a$:

If $a=1$, then there are $3$ cases:

  • $b=2,c=50$
  • $b=4,c=25$
  • $b=5,c=20$

If $a=2$, then there is only $1$ case:

  • $b=5,c=10$

In total, there are $3+1=\boxed{\textbf{(E) } 4}$ ways to choose distinct positive integer values of $a,b,c$.

~MathFun1000

Video Solution 1

https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=142

Video Solution 2

https://youtu.be/LHnC_Wz6fOU

Video Solution

https://youtu.be/Q0R6dnIO95Y?t=98

See Also

2022 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
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 AJHSME/AMC 8 Problems and Solutions

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