Difference between revisions of "2012 UNCO Math Contest II Problems/Problem 1"
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== Solution == | == Solution == | ||
(a) <math>36</math> (b) <math>196</math> | (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 == | ||
Latest revision as of 20:52, 8 January 2024
Problem
(a) What is the largest factor of 
that is not a multiple of 
?
(b) If 
satisfies 
,
then what is the largest perfect square that is a factor of ?
Solution
(a)(b)
(b) 
The prime factorization of is
.
We need to get the largest factor of which isn't a multiple of , so we can't include both 
and 
. 
is greater than , so we'll use that and 
to get 
.
The largest square factor of is 
= 
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 | ||
