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Difference between revisions of "2003 AMC 8 Problems/Problem 19"

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<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math>
 
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math>
  
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==Solution==
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==Solution 1==
  
 
Find the least common multiple of <math>15, 20, 25</math> by turning the numbers into their prime factorization. <cmath>15 = 3 * 5, 20 = 2^2 * 5, 25 = 5^2</cmath> Gather all necessary multiples  
 
Find the least common multiple of <math>15, 20, 25</math> by turning the numbers into their prime factorization. <cmath>15 = 3 * 5, 20 = 2^2 * 5, 25 = 5^2</cmath> Gather all necessary multiples  
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Notice that <math>1000</math> can be prime factorized into:
 
Notice that <math>1000</math> can be prime factorized into:
 
<cmath>1000 = 2 * 2 * 2 * 5 * 5 * 5</cmath>
 
<cmath>1000 = 2 * 2 * 2 * 5 * 5 * 5</cmath>
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Using this, we can remove all the common factors of <math>15, 20,</math> and <math>25</math> that are shared with <math>1000</math>:
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Now take the lowest common multiple of <math>15,20</math> and <math>25</math> which is <math>300</math>:
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<cmath> 3 * 5 * \cancel{2} * \cancel{2} * \cancel{5} * \cancel{5} * \cancel{5}</cmath>
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<cmath>\text{LCM}(15,20,25) = 300</cmath>
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We must also cancel the same factors in <math>1000</math> and <math>2000</math> to ensure that we don't exceed our range:
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Using <math>300</math>'s prime factorization, we can cancel the following factors that are common in both <math>300</math> and <math>1000</math>:
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<cmath>1000 = 2 * \cancel{2} * \cancel{2} * \cancel{5} * \cancel{5} * \cancel{5}</cmath>
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<cmath> 300 = 3 * \cancel{2} * \cancel{2} * \cancel{5} * \cancel{5}</cmath>
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<cmath>1000 = 2 * 5 * \cancel{2} * \cancel{2} * \cancel{5} * \cancel{5}</cmath>
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<cmath>2000 = 2 * 2 * 5 * \cancel{2} * \cancel{2} * \cancel{5} * \cancel{5}</cmath>
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We now get the following factors from both of them:
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The product of the remaining factors of <math>1000</math> is <math>2</math>, while the product of the remaining factors of <math>2000</math> is <math>20</math>.
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<cmath>3, 2, \text{and } 5</cmath>
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The remaining numbers left of <math>15, 20</math>, and <math>25</math> (<math>3</math> and <math>5</math>) yield:
 
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<cmath>3, 5, 15</cmath>
 
 
Thus, counting these numbers we get our answer of: <math>\boxed{\textbf{(C)}\ 3}</math>.
 
Thus, counting these numbers we get our answer of: <math>\boxed{\textbf{(C)}\ 3}</math>.
  
  
 
~Hawk2019
 
~Hawk2019
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(Note that <math>3, 5,</math> and <math>15</math> are all less than <math>20</math>, but greater than <math>2</math>. Had they been larger than <math>20</math> or less than <math>2</math>, they wouldn't have been between <math>1000</math> and <math>2000</math>)
 
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2003|num-b=18|num-a=20}}
 
{{AMC8 box|year=2003|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:32, 29 July 2024

Problem

How many integers between 1000 and 2000 have all three of the numbers 15, 20, and 25 as factors?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Solution 1

Find the least common multiple of $15, 20, 25$ by turning the numbers into their prime factorization. \[15 = 3 * 5, 20 = 2^2 * 5, 25 = 5^2\] Gather all necessary multiples $3, 2^2, 5^2$ when multiplied gets $300$. The multiples of $300 - 300, 600, 900, 1200, 1500, 1800, 2100$. The number of multiples between 1000 and 2000 is $\boxed{\textbf{(C)}\ 3}$.

Solution 2

Using the previous solution, turn $15, 20,$ and $25$ into their prime factorizations. \[15 = 3 * 5, 20 = 2^2 * 5, 25 = 5^2\] Notice that $1000$ can be prime factorized into: \[1000 = 2 * 2 * 2 * 5 * 5 * 5\] Now take the lowest common multiple of $15,20$ and $25$ which is $300$: \[\text{LCM}(15,20,25) = 300\] Using $300$'s prime factorization, we can cancel the following factors that are common in both $300$ and $1000$: \[300 = 3 * \cancel{2} * \cancel{2} * \cancel{5} * \cancel{5}\] \[1000 = 2 * 5 * \cancel{2} * \cancel{2} * \cancel{5} * \cancel{5}\] We now get the following factors from both of them: \[3, 2, \text{and } 5\] Thus, counting these numbers we get our answer of: $\boxed{\textbf{(C)}\ 3}$.


~Hawk2019

See Also

2003 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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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