Difference between revisions of "2019 AIME I Problems/Problem 7"
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==Solution 1== | ==Solution 1== | ||
− | Add the two equations to get that <math>\log x+\log y+2(\log(\gcd(x,y))+\log(\text{lcm}(x,y)))=630</math>. | + | Add the two equations to get that <math>\log x+\log y+2(\log(\gcd(x,y))+2(\log(\text{lcm}(x,y)))=630</math>. |
Then, we use the theorem <math>\log a+\log b=\log ab</math> to get the equation, <math>\log (xy)+2(\log(\gcd(x,y))+\log(\text{lcm}(x,y)))=630</math>. | Then, we use the theorem <math>\log a+\log b=\log ab</math> to get the equation, <math>\log (xy)+2(\log(\gcd(x,y))+\log(\text{lcm}(x,y)))=630</math>. | ||
Using the theorem that <math>\gcd(x,y) \cdot \text{lcm}(x,y)=x\cdot y</math>, along with the previously mentioned theorem, we can get the equation <math>3\log(xy)=630</math>. | Using the theorem that <math>\gcd(x,y) \cdot \text{lcm}(x,y)=x\cdot y</math>, along with the previously mentioned theorem, we can get the equation <math>3\log(xy)=630</math>. | ||
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<math>10^{210}</math> can be factored into <math>2^{210} \cdot 5^{210}</math>, and <math>m+n</math> equals to the sum of the exponents of <math>2</math> and <math>5</math>, which is <math>210+210 = 420</math>. | <math>10^{210}</math> can be factored into <math>2^{210} \cdot 5^{210}</math>, and <math>m+n</math> equals to the sum of the exponents of <math>2</math> and <math>5</math>, which is <math>210+210 = 420</math>. | ||
Multiply by two to get <math>2m +2n</math>, which is <math>840</math>. | Multiply by two to get <math>2m +2n</math>, which is <math>840</math>. | ||
− | Then, use the first equation (<math>\log x + 2\log(\gcd(x,y)) = 60</math>) to show that <math>x</math> has to have lower degrees of <math>2</math> and <math>5</math> than <math>y</math> (you can also test when <math>x>y</math>, which is a contradiction to the restrains | + | Then, use the first equation (<math>\log x + 2\log(\gcd(x,y)) = 60</math>) to show that <math>x</math> has to have lower degrees of <math>2</math> and <math>5</math> than <math>y</math> (you can also test when <math>x>y</math>, which is a contradiction to the restrains we set before). Therefore, <math>\gcd(x,y)=x</math>. Then, turn the equation into <math>3\log x = 60</math>, which yields <math>\log x = 20</math>, or <math>x = 10^{20}</math>. |
Factor this into <math>2^{20} \cdot 5^{20}</math>, and add the two 20's, resulting in <math>m</math>, which is <math>40</math>. | Factor this into <math>2^{20} \cdot 5^{20}</math>, and add the two 20's, resulting in <math>m</math>, which is <math>40</math>. | ||
Add <math>m</math> to <math>2m + 2n</math> (which is <math>840</math>) to get <math>40+840 = \boxed{880}</math>. | Add <math>m</math> to <math>2m + 2n</math> (which is <math>840</math>) to get <math>40+840 = \boxed{880}</math>. | ||
− | == | + | ~minor mistake fix by virjoy2001 |
+ | ~minor mistake fix by oralayhan | ||
+ | |||
+ | Remark: You can obtain the contradiction by using LTE. If <math>\nu_2{(x)}\geq{\nu_2{(y)}}, \nu_2{(y^2x)}=60</math>. However, <math>\nu_2{(xy)}=210</math> a contradiction. Same goes with taking <math>\nu_5{(x,y)}</math> | ||
+ | |||
+ | ===Easier Approach to Finish=== | ||
+ | After noting that <math>xy=10^{210},</math> notice that we can let <math>x=10^a</math> and <math>y=10^b.</math> Thus, we have from the given equations (1) and (2) respectively, that: | ||
+ | <cmath>a+2a=60</cmath> | ||
+ | <cmath>b+2b=570</cmath> | ||
+ | Solving, we get <math>(a,b)=(20,190).</math> This matches with our constraint that <math>xy=10^{210}</math> (this constraint can actually be rederived by adding the two equations) so we finish from here. | ||
+ | |||
+ | <math>x=2^{20}\cdot 5^{20}, y=2^{190}\cdot 5^{190}.</math> Thus, the answer desired is <math>3(20+20)+2(190+190)=880.</math> ~mathboy282 (minor addition by Technodoggo) | ||
+ | |||
+ | ==Solution 2== | ||
First simplifying the first and second equations, we get that | First simplifying the first and second equations, we get that | ||
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==Solution 3 (Easy Solution)== | ==Solution 3 (Easy Solution)== | ||
Let <math>x=10^a</math> and <math>y=10^b</math> and <math>a<b</math>. Then the given equations become <math>3a=60</math> and <math>3b=570</math>. Therefore, <math>x=10^{20}=2^{20}\cdot5^{20}</math> and <math>y=10^{190}=2^{190}\cdot5^{190}</math>. Our answer is <math>3(20+20)+2(190+190)=\boxed{880}</math>. | Let <math>x=10^a</math> and <math>y=10^b</math> and <math>a<b</math>. Then the given equations become <math>3a=60</math> and <math>3b=570</math>. Therefore, <math>x=10^{20}=2^{20}\cdot5^{20}</math> and <math>y=10^{190}=2^{190}\cdot5^{190}</math>. Our answer is <math>3(20+20)+2(190+190)=\boxed{880}</math>. | ||
+ | |||
==Solution 4 == | ==Solution 4 == | ||
We will use the notation <math>(a, b)</math> for <math>\gcd(a, b)</math> and <math>[a, b]</math> as <math>\text{lcm}(a, b)</math>. | We will use the notation <math>(a, b)</math> for <math>\gcd(a, b)</math> and <math>[a, b]</math> as <math>\text{lcm}(a, b)</math>. | ||
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(Solution by Prabh1512) | (Solution by Prabh1512) | ||
==Solution 6 (Official MAA)== | ==Solution 6 (Official MAA)== | ||
− | The two equations are equivalent to <math>x(\gcd(x,y))^2=10^{60}</math> and <math>y(\operatorname{lcm}(x,y))^2=10^{570}</math> respectively. Multiplying corresponding sides of the equations leads to <math>xy(\gcd(x,y)\operatorname{lcm}(x,y))^2=(xy)^3=10^{630}</math>, so <math>xy=10^{210}</math>. It follows that there are nonnegative integers <math>a,\,b,\,c,</math> and <math>d</math> such that <math>(x,y)=(2^a5^b,2^c5^d)</math> with <math>a+c=b+d=210</math>. Furthermore, <cmath>\frac{(\operatorname{lcm}(x,y))^2}{x}=\frac{( | + | The two equations are equivalent to <math>x(\gcd(x,y))^2=10^{60}</math> and <math>y(\operatorname{lcm}(x,y))^2=10^{570}</math> respectively. Multiplying corresponding sides of the equations leads to <math>xy(\gcd(x,y)\operatorname{lcm}(x,y))^2=(xy)^3=10^{630}</math>, so <math>xy=10^{210}</math>. It follows that there are nonnegative integers <math>a,\,b,\,c,</math> and <math>d</math> such that <math>(x,y)=(2^a5^b,2^c5^d)</math> with <math>a+c=b+d=210</math>. Furthermore, <cmath>\frac{(\operatorname{lcm}(x,y))^2}{x}=\frac{y(\operatorname{lcm}(x,y))^2}{xy}=\frac{10^{570}}{10^{210}}=10^{360}.</cmath> Thus <math>\max(2a,2c)-a=\max(2b,2d)-b=360.</math> Because neither <math>2a-a</math> nor <math>2b-b</math> can equal <math>360</math> when <math>a+c=b+d=210,</math> it follows that <math>2c-a=2d-b=360</math>. Hence <math>(a,b,c,d)=(20,20,190,190</math>, so the prime factorization of <math>x</math> has <math>20+20=40</math> prime factors, and the prime factorization of <math>y</math> has <math>190+190=380</math> prime factors. The requested sum is <math>3\cdot40+2\cdot380=880.</math> |
+ | |||
+ | ==Solution 7== | ||
+ | Add the two equations and use the fact that <math>\gcd\left(x,y\right)\cdot\mathrm{lcm}\left(x,y\right)=xy</math> to find that <math>xy=10^{210}</math>. So let <math>x=2^a5^b</math> and <math>y=2^{210-a}5^{210-b}</math> for <math>0\leq a,b\leq210</math>. If <math>a\geq105</math> then the exponent of <math>2</math> in <math>x\cdot\gcd\left(x,y\right)^2=10^{60}</math> is <math>a+2\left(210-a\right)=420-a</math>, so <math>a=360</math>, contradiction. So <math>a<105</math>. Then the exponent of <math>2</math> in <math>x\cdot\gcd\left(x,y\right)^2</math> is <math>a+2a=3a</math>, so <math>a=20</math>. Similarly, <math>b=20</math>. Then <math>3m+2n=3\left(a+b\right)+2\left(420-a-b\right)=\boxed{880}</math> as desired. | ||
+ | |||
+ | ~from trumpeter in the AoPS Forums Contest Discussion | ||
+ | |||
+ | ==Solution 8== | ||
+ | We can simplify the equations step by step. The first equation simplifies to <math>log(</math>(x)(<math>(gcd(x,y))^2</math>)<math>)=60</math>. The second equation simplifies to log(<math>(y)</math>(<math>(lcm(x,y)^2</math>)<math>)=570</math>. Up to here, we used the exponent and addition log identities. | ||
+ | |||
+ | Now before we move on to the next few simplification steps, we must remember that <math>gcd(a,b)</math>*<math>lcm(a,b)</math>=<math>a*b</math>. | ||
+ | |||
+ | Due to the fact that these log's are in base <math>10</math>, this makes the first and second equation equal to <math>10^{60}</math>, <math>10^{570}</math> respectively. In this step, we switched the log's into exponential form. Now we multiply both equations to get <math>x*y</math>*<math>(xy)^2</math>=(<math>x^3</math>)(<math>y^3</math>)=<math>10^{630}</math>. Now we take the cube root of both sides to get <math>xy=10^{210}</math>. | ||
+ | |||
+ | We've now gotten to the crucial part of this equation. Though this wouldn't pass for full points in a proof-based contest, this is AIME. So, we assume that <math>x<y</math>. We also let <math>x</math>=<math>10^a</math> and <math>y</math>=<math>10^b</math> That means that <math>gcd(x,y)</math> is <math>x</math> and the <math>lcm(x,y)</math> is <math>y</math> due to the fact that we are also assuming that both <math>x,y</math> are <math>10^a</math>, <math>10^b</math> respectively. | ||
+ | |||
+ | If we put our last few insights together into the first and second equation, we see that <math>x</math>=<math>10^{20}</math>. We also see that <math>y</math>=<math>10^{190}</math>. We could check these if wanted (don't worry they work), but if you were very limited on time for this question, just assume these values work and move on. | ||
+ | |||
+ | Now <math>10^{20}</math> factors as <math>2^{20}</math>*<math>5^{20}</math>. This has <math>40</math> prime factors. <math>2</math>, <math>20</math> times and <math>5</math>, <math>20</math> times. <math>10^{190}</math> factors as <math>2^{190}</math>*<math>5^{190}</math>. This has <math>380</math> prime factors. <math>2</math>, <math>190</math> times and <math>5</math>, <math>190</math> times. Now it's just <math>40*3+380*2=880</math> as our final answer. | ||
+ | |||
+ | -Schintalpati | ||
+ | |||
+ | ==Video Solution(Pretty Straightforward) == | ||
+ | https://www.youtube.com/watch?v=NOLk9-A4eDo | ||
+ | Remember to subscribe! | ||
+ | |||
+ | ~North America math Contest Go Go Go | ||
==See Also== | ==See Also== | ||
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[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
+ | [[Category:Intermediate Algebra Problems]] | ||
+ | |||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 00:44, 23 October 2024
Contents
Problem
There are positive integers and that satisfy the system of equations Let be the number of (not necessarily distinct) prime factors in the prime factorization of , and let be the number of (not necessarily distinct) prime factors in the prime factorization of . Find .
Solution 1
Add the two equations to get that . Then, we use the theorem to get the equation, . Using the theorem that , along with the previously mentioned theorem, we can get the equation . This can easily be simplified to , or .
can be factored into , and equals to the sum of the exponents of and , which is . Multiply by two to get , which is . Then, use the first equation () to show that has to have lower degrees of and than (you can also test when , which is a contradiction to the restrains we set before). Therefore, . Then, turn the equation into , which yields , or . Factor this into , and add the two 20's, resulting in , which is . Add to (which is ) to get .
~minor mistake fix by virjoy2001 ~minor mistake fix by oralayhan
Remark: You can obtain the contradiction by using LTE. If . However, a contradiction. Same goes with taking
Easier Approach to Finish
After noting that notice that we can let and Thus, we have from the given equations (1) and (2) respectively, that: Solving, we get This matches with our constraint that (this constraint can actually be rederived by adding the two equations) so we finish from here.
Thus, the answer desired is ~mathboy282 (minor addition by Technodoggo)
Solution 2
First simplifying the first and second equations, we get that
Thus, when the two equations are added, we have that
When simplified, this equals
so this means that
so
Now, the following cannot be done on a proof contest but let's (intuitively) assume that and and are both powers of . This means the first equation would simplify to and Therefore, and and if we plug these values back, it works! has total factors and has so
Please remember that you should only assume on these math contests because they are timed; this would technically not be a valid solution.
Solution 3 (Easy Solution)
Let and and . Then the given equations become and . Therefore, and . Our answer is .
Solution 4
We will use the notation for and as . We can start with a similar way to Solution 1. We have, by logarithm properties, or . We can do something similar to the second equation and our two equations become Adding the two equations gives us . Since we know that , , or . We can express as and as . Another way to express is now , and is now . We know that , and thus, , and . Our equations for and now become or . Doing the same for the equation, we have , and , which satisfies . Thus, . ~awsomek
Solution 5
Let . Simplifying, . Notice that since are coprime, and (Prove it yourself !) , . Hence, giving the answer .
(Solution by Prabh1512)
Solution 6 (Official MAA)
The two equations are equivalent to and respectively. Multiplying corresponding sides of the equations leads to , so . It follows that there are nonnegative integers and such that with . Furthermore, Thus Because neither nor can equal when it follows that . Hence , so the prime factorization of has prime factors, and the prime factorization of has prime factors. The requested sum is
Solution 7
Add the two equations and use the fact that to find that . So let and for . If then the exponent of in is , so , contradiction. So . Then the exponent of in is , so . Similarly, . Then as desired.
~from trumpeter in the AoPS Forums Contest Discussion
Solution 8
We can simplify the equations step by step. The first equation simplifies to (x)(). The second equation simplifies to log((). Up to here, we used the exponent and addition log identities.
Now before we move on to the next few simplification steps, we must remember that *=.
Due to the fact that these log's are in base , this makes the first and second equation equal to , respectively. In this step, we switched the log's into exponential form. Now we multiply both equations to get *=()()=. Now we take the cube root of both sides to get .
We've now gotten to the crucial part of this equation. Though this wouldn't pass for full points in a proof-based contest, this is AIME. So, we assume that . We also let = and = That means that is and the is due to the fact that we are also assuming that both are , respectively.
If we put our last few insights together into the first and second equation, we see that =. We also see that =. We could check these if wanted (don't worry they work), but if you were very limited on time for this question, just assume these values work and move on.
Now factors as *. This has prime factors. , times and , times. factors as *. This has prime factors. , times and , times. Now it's just as our final answer.
-Schintalpati
Video Solution(Pretty Straightforward)
https://www.youtube.com/watch?v=NOLk9-A4eDo Remember to subscribe!
~North America math Contest Go Go Go
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.