Difference between revisions of "2016 AMC 10A Problems/Problem 25"
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− | It is well known that if the <math>\text{lcm}a,b=c</math> and <math>c</math> can be written as <math>p_1^ap_2^bp_3^c\dots</math>, then the highest power of all prime numbers <math>p_1,p_2,p_3\dots</math> must divide into either <math>a</math> and/or <math>b</math>. Or else a lower <math>c_0=p_1^{a-\epsilon}p_2^{b-\epsilon}p_3^{c-\epsilon}\dots</math> is the <math>\text{lcm}</math>. | + | It is well known that if the <math>\text{lcm}(a,b)=c</math> and <math>c</math> can be written as <math>p_1^ap_2^bp_3^c\dots</math>, then the highest power of all prime numbers <math>p_1,p_2,p_3\dots</math> must divide into either <math>a</math> and/or <math>b</math>. Or else a lower <math>c_0=p_1^{a-\epsilon}p_2^{b-\epsilon}p_3^{c-\epsilon}\dots</math> is the <math>\text{lcm}</math>. |
Start from <math>x</math>:<math>\text{lcm}(x,y)=72</math> so <math>8\mid x</math> or <math>9\mid x</math> or both. But <math>9\nmid x</math> because <math>\text{lcm}(x,z}=600</math> and <math>9\nmid 600</math>. | Start from <math>x</math>:<math>\text{lcm}(x,y)=72</math> so <math>8\mid x</math> or <math>9\mid x</math> or both. But <math>9\nmid x</math> because <math>\text{lcm}(x,z}=600</math> and <math>9\nmid 600</math>. |
Revision as of 22:18, 5 February 2016
Contents
Problem
How many ordered triples of positive integers satisfy and ?
Solution 1
We prime factorize and . The prime factorizations are , and , respectively. Let , and . We know that and since isn't a multiple of 5. Since we know that . We also know that since that . So now some equations have become useless to us...let's take them out. are the only two important ones left. We do casework on each now. If then or . Similarly if then . Thus our answer is .
Solution 2
It is well known that if the and can be written as , then the highest power of all prime numbers must divide into either and/or . Or else a lower is the .
Start from : so or or both. But because $\text{lcm}(x,z}=600$ (Error compiling LaTeX. ! Extra }, or forgotten $.) and . So .
can be in both cases of but NOT because $\lcm{y,z}=900$ (Error compiling LaTeX. ! Undefined control sequence.) and .
So there are six sets of and we will list all possible values of based on those.
because must source all powers of . . $z\nin\{200,225\}$ (Error compiling LaTeX. ! Undefined control sequence.) because of restrictions.
By different sourcing of powers of and ,
Counting the cases,
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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 AMC 10 Problems and Solutions |
2016 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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 AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.