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Difference between revisions of "2018 AMC 8 Problems/Problem 14"
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==Problem 14==
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==Problem==
 
Let <math>N</math> be the greatest five-digit number whose digits have a product of <math>120</math>. What is the sum of the digits of <math>N</math>?
 
Let <math>N</math> be the greatest five-digit number whose digits have a product of <math>120</math>. What is the sum of the digits of <math>N</math>?
  
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== Solution ==  
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<math>\textbf{(A) }15\qquad\textbf{(B) }16\qquad\textbf{(C) }17\qquad\textbf{(D) }18\qquad\textbf{(E) }20</math>
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If we start off with the first digit, we know that it can't by <math>9</math> since <math>9</math> is not a factor of <math>120</math>. We scale down to the digit <math>8</math>, which does work since it is a factor of <math>120</math>. Now, we have to know what digits will take up the remaining four spots. To find this result, just divide <math>\frac{120}{8}=15</math>. The next place can be <math>5</math>, as it is the largest factor, aside from <math>15</math>. Consequently, our next three values will be <math>3,1</math> and <math>1</math> if we use the same logic! Therefore, our five-digit number is <math>85311</math>, so the sum is <math>8+5+3+1+1=\boxed{18}, \textbf{(D)}</math> -mathmaster010
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== Solution 1==  
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If we start off with the first digit, we know that it can't be <math>9</math> since <math>9</math> is not a factor of <math>120</math>. We go down to the digit <math>8</math>, which does work since it is a factor of <math>120</math>. Now, we have to know what digits will take up the remaining four spots. To find this result, just divide <math>\frac{120}{8}=15</math>. The next place can be <math>5</math>, as it is the largest factor, aside from <math>15</math>. Consequently, our next three values will be <math>3,1</math> and <math>1</math> if we use the same logic. Therefore, our five-digit number is <math>85311</math>, so the sum is <math>8+5+3+1+1=18\implies \boxed{\textbf{(D) }18}</math>.
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== Solution 2 (Factorial) ==
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<math>120</math> is <math>5!</math>, so we have <math>(5)(4)(3)(2)(1) = 120</math>. (Alternatively, you could identify the prime factors <math>(5)(3)(2)(2)(2) = 120</math>.) Now look for the largest digit you can create by combining these factors.
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<math>8=4 \cdot 2</math>
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Use this largest digit for the ten-thousands place: <math>8</math>_ , _ _ _
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Next you use the <math>5</math> and the <math>3</math> for the next places: <math>85,3</math> _ _ (You can't use <math>3 \cdot 2=6</math> because the <math>2</math> was used to make <math>8</math>.)
  
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Fill the remaining places with 1: <math>85,311</math>
  
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<math> = (5)(8)(3)(1)(1) =120</math>
  
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<math>\textbf{(A) }15\qquad\textbf{(B) }16\qquad\textbf{(C) }17\qquad\textbf{(D) }18\qquad\textbf{(E) }20</math>
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<math>8+5+3+1+1=\boxed{\textbf{(D) }18}</math>.
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==Video Solution (CREATIVE ANALYSIS!!!)==
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https://youtu.be/zxEO6amczPU
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~Education, the Study of Everything
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==Video Solutions==
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https://youtu.be/IAKhC_A0kok
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https://youtu.be/7an5wU9Q5hk?t=13
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https://youtu.be/Q5YrDW62VDU
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~savannahsolver
  
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==See Also==
 
{{AMC8 box|year=2018|num-b=13|num-a=15}}
 
{{AMC8 box|year=2018|num-b=13|num-a=15}}
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{{MAA Notice}}

Latest revision as of 10:01, 10 April 2024

Problem

Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$?

$\textbf{(A) }15\qquad\textbf{(B) }16\qquad\textbf{(C) }17\qquad\textbf{(D) }18\qquad\textbf{(E) }20$

Solution 1

If we start off with the first digit, we know that it can't be $9$ since $9$ is not a factor of $120$. We go down to the digit $8$, which does work since it is a factor of $120$. Now, we have to know what digits will take up the remaining four spots. To find this result, just divide $\frac{120}{8}=15$. The next place can be $5$, as it is the largest factor, aside from $15$. Consequently, our next three values will be $3,1$ and $1$ if we use the same logic. Therefore, our five-digit number is $85311$, so the sum is $8+5+3+1+1=18\implies \boxed{\textbf{(D) }18}$.

Solution 2 (Factorial)

$120$ is $5!$, so we have $(5)(4)(3)(2)(1) = 120$. (Alternatively, you could identify the prime factors $(5)(3)(2)(2)(2) = 120$.) Now look for the largest digit you can create by combining these factors.

$8=4 \cdot 2$

Use this largest digit for the ten-thousands place: $8$_ , _ _ _

Next you use the $5$ and the $3$ for the next places: $85,3$ _ _ (You can't use $3 \cdot 2=6$ because the $2$ was used to make $8$.)

Fill the remaining places with 1: $85,311$

$= (5)(8)(3)(1)(1) =120$

$8+5+3+1+1=\boxed{\textbf{(D) }18}$.

Video Solution (CREATIVE ANALYSIS!!!)

https://youtu.be/zxEO6amczPU

~Education, the Study of Everything

Video Solutions

https://youtu.be/IAKhC_A0kok

https://youtu.be/7an5wU9Q5hk?t=13

https://youtu.be/Q5YrDW62VDU

~savannahsolver

See Also

2018 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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All AJHSME/AMC 8 Problems and Solutions

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