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

Revision as of 22:39, 8 January 2024

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$ (Error compiling LaTeX. Unknown error_msg)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 by Math-X (First understand the problem!!!)

https://youtu.be/oUEa7AjMF2A?si=tkBYOey2NioTPPPq&t=221

~Math-X

Video Solution 2 (CREATIVE THINKING!!!)

https://youtu.be/5-6zj2mBBSA

~Education, the Study of Everything

Video Solution 3

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

~Interstigation

Video Solution 4

https://youtu.be/LHnC_Wz6fOU

~savannahsolver

Video Solution 5

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

~STEMbreezy

Video Solution 6

https://www.youtube.com/watch?v=KkZ95iNlFyc

~harungurcan

@@ Line 14: / Line 14: @@
 * If <math>c=20,</math> then <math>(a,b,c)=(1,5,20).</math>
-* If <math>c=25,</math> then <math>(a,b,c)=(1,4,25).</math>
+* If <math>c=25,</math> then <math>(a,b,c)=(1,4,25).
-* If <math>c=50,</math> then <math>(a,b,c)=(1,2,50).</math>
+* If </math>c=50,<math> then </math>(a,b,c)=(1,2,50).<math>
-Together, the numbers <math>a,b,</math> and <math>c</math> can be chosen in <math>\boxed{\textbf{(E) } 4}</math> ways.
+Together, the numbers </math>a,b,<math> and </math>c<math> can be chosen in </math>\boxed{\textbf{(E) } 4}$ ways.
 ~MRENTHUSIASM

2022 AMC 8 (Problems • Answer Key • Resources)
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