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Difference between revisions of "2012 UNCO Math Contest II Problems/Problem 1"

(Created page with "== Problem == (a) What is the largest factor of <math>180</math> that is not a multiple of <math>15</math>? (b) If <math>N</math> satisfies <math>2^5 \times 3^2 \times5^4 \time...")
 
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== Solution ==
 
== Solution ==
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(a) <math>36</math> (b) <math>196</math>
  
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<math>(a)</math>
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The prime factorization of <math>180</math> is
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<math>2^2 \times 3^2 \times5</math>.
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We need to get the largest factor of <math>180</math> which isn't a multiple of <math>15</math>, so we can't include both <math>3</math> and <math>5</math>. <math>3^2</math> is greater than <math>5</math>, so we'll use that and <math>2^2</math> to get <math>3 \times 3 \times 2 \times 2 = \fbox{\textbf 36}</math>.
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<math>(b)</math>
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<math>2^5 \times 3^2 \times5^4 \times 7^3 \times 11=99000 \times N</math>
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<math>2^5 \times 3^2 \times5^4 \times 7^3 \times 11=2^3 \times 3^2 \times 5^3 \times 11 \times N</math>
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<math>2^2 \times 5 \times 7^3= N</math>
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The largest square factor of <math>N</math> is <math>2^2</math> <math>\times</math> <math>7^2</math> = <math>\fbox{\textbf 196}</math>
  
 
== See Also ==
 
== See Also ==
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{{UNC Math Contest box|n=II|year=2012|before=First Problem|num-a=2}}
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{{UNCO Math Contest box|n=II|year=2012|before=First Problem|num-a=2}}
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]

Latest revision as of 20:52, 8 January 2024

Problem

(a) What is the largest factor of $180$ that is not a multiple of $15$?

(b) If $N$ satisfies $2^5 \times 3^2 \times5^4 \times 7^3 \times 11=99000 \times N$, then what is the largest perfect square that is a factor of $N$?


Solution

(a) $36$ (b) $196$

$(a)$


The prime factorization of $180$ is


$2^2 \times 3^2 \times5$.


We need to get the largest factor of $180$ which isn't a multiple of $15$, so we can't include both $3$ and $5$. $3^2$ is greater than $5$, so we'll use that and $2^2$ to get $3 \times 3 \times 2 \times 2 = \fbox{\textbf 36}$.


$(b)$


$2^5 \times 3^2 \times5^4 \times 7^3 \times 11=99000 \times N$


$2^5 \times 3^2 \times5^4 \times 7^3 \times 11=2^3 \times 3^2 \times 5^3 \times 11 \times N$


$2^2 \times 5 \times 7^3= N$


The largest square factor of $N$ is $2^2$ $\times$ $7^2$ = $\fbox{\textbf 196}$

See Also

2012 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
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