Difference between revisions of "2019 AIME I Problems/Problem 7"

Revision as of 19:11, 31 May 2020

Problem 7

There are positive integers $x$ and $y$ that satisfy the system of equations $\begin{align*} \log_{10} x + 2 \log_{10} (\text{gcd}(x,y)) &= 60\\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*}$ Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$ , and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y$ . Find $3m+2n$ .

Solution 1

Add the two equations to get that $\log x+\log y+2(\log(\gcd(x,y))+\log(\text{lcm}(x,y)))=630$ . Then, we use the theorem $\log a+\log b=\log ab$ to get the equation, $\log (xy)+2(\log(\gcd(x,y))+\log(\text{lcm}(x,y)))=630$ . Using the theorem that $\gcd(x,y) \cdot \text{lcm}(x,y)=x\cdot y$ , along with the previously mentioned theorem, we can get the equation $3\log(xy)=630$ . This can easily be simplified to $\log(xy)=210$ , or $xy = 10^{210}$ .

$10^{210}$ can be factored into $2^{210} \cdot 5^{210}$ , and $m+n$ equals to the sum of the exponents of $2$ and $5$ , which is $210+210 = 420$ . Multiply by two to get $2m +2n$ , which is $840$ . Then, use the first equation ( $\log x + 2\log(\gcd(x,y)) = 60$ ) to show that $x$ has to have lower degrees of $2$ and $5$ than $y$ (you can also test when $x>y$ , which is a contradiction to the restrains you set before). Therefore, making the $\gcd(x)$ . Then, turn the equation into $3\log x = 60$ , which yields $\log x = 20$ , or $x = 10^{20}$ . Factor this into $2^{20} \cdot 5^{20}$ , and add the two 20's, resulting in $m$ , which is $40$ . Add $m$ to $2m + 2n$ (which is $840$ ) to get $40+840 = \boxed{880}$ .

Solution 2 (Crappier Solution)

First simplifying the first and second equations, we get that

$\log_{10}(x\cdot\text{gcd}(x,y)^2)=60$ $\log_{10}(y\cdot\text{lcm}(x,y)^2)=570$

Thus, when the two equations are added, we have that $\log_{10}(x\cdot y\cdot\text{gcd}^2\cdot\text{lcm}^2)=630$ When simplified, this equals $\log_{10}(x^3y^3)=630$ so this means that $x^3y^3=10^{630}$ so $xy=10^{210}.$

Now, the following cannot be done on a proof contest but let's (intuitively) assume that $x<y$ and $x$ and $y$ are both powers of $10$ . This means the first equation would simplify to $x^3=10^{60}$ and $y^3=10^{570}.$ Therefore, $x=10^{20}$ and $y=10^{190}$ and if we plug these values back, it works! $10^{20}$ has $20\cdot2=40$ total factors and $10^{190}$ has $190\cdot2=380$ so $3\cdot 40 + 2\cdot 380 = \boxed{880}.$

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 $x=10^a$ and $y=10^b$ and $a<b$ . Then the given equations become $3a=60$ and $3b=570$ . Therefore, $x=10^{20}=2^{20}\cdot5^{20}$ and $y=10^{190}=2^{190}\cdot5^{190}$ . Our answer is $3(20+20)+2(190+190)=\boxed{880}$ .

Solution 4

We will use the notation $(a, b)$ for $\gcd(a, b)$ and $[a, b]$ as $\text{lcm}(a, b)$ . We can start with a similar way to Solution 1. We have, by logarithm properties, $\log_{10}{x}+\log_{10}{(x, y)^2}=60$ or $x(x, y)^2=10^{60}$ . We can do something similar to the second equation and our two equations become $x(x, y)^2=10^{60}$ $y[x, y]^2=10^{570}$ Adding the two equations gives us $xy(x, y)^2[x, y]^2=10^{630}$ . Since we know that $(a, b)\cdot[a, b]=ab$ , $x^3y^3=10^{630}$ , or $xy=10^{210}$ . We can express $x$ as $2^a5^b$ and $y$ as $2^c5^d$ . Another way to express $(x, y)$ is now $2^{min(a, c)}5^{min(b, d)}$ , and $[x, y]$ is now $2^{max(a, c)}5^{max(b, d)}$ . We know that $x<y$ , and thus, $a<c$ , and $b<d$ . Our equations for $lcm$ and $gcd$ now become $2^a5^b(2^a5^a)^2=10^{60}$ or $a=b=20$ . Doing the same for the $lcm$ equation, we have $c=d=190$ , and $190+20=210$ , which satisfies $xy=210$ . Thus, $3m+2n=3(20+20)+2(190+190)=\boxed{880}$ . ~awsomek

@@ Line 14: / Line 14: @@
 This can easily be simplified to <math>\log(xy)=210</math>, or <math>xy = 10^{210}</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 2 and 5, 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>.
-Then, use the first equation (<math>\log x + 2\log(\gcd(x,y)) = 60</math>) to show that x has to have lower degrees of 2 and 5 than y. Therefore, making the <math>\gcd(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>.
+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 you set before). Therefore, making the <math>\gcd(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 m, which is 40.
+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 840) 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>.
 ==Solution 2 (Crappier Solution)==

2019 AIME I (Problems • Answer Key • Resources)
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