Difference between revisions of "2003 AMC 12B Problems/Problem 18"
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<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34</math> | <math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34</math> | ||
− | == Solution == | + | == Solution 1== |
Substitute <math>a^cb^d</math> into <math>x</math>. We then have <math>7(a^{5c}b^{5d}) = 11y^{13}</math>. Divide both sides by <math>7</math>, and it follows that: | Substitute <math>a^cb^d</math> into <math>x</math>. We then have <math>7(a^{5c}b^{5d}) = 11y^{13}</math>. Divide both sides by <math>7</math>, and it follows that: | ||
− | <cmath>(a^{5c}b^{5d}) = | + | <cmath>(a^{5c}b^{5d}) = \frac{11y^{13}}{7}.</cmath> |
Note that because <math>11</math> and <math>7</math> are prime, the minimum value of <math>x</math> must involve factors of <math>7</math> and <math>11</math> only. Thus, we try to look for the lowest power <math>p</math> of <math>11</math> such that <math>13p + 1 \equiv 0 \pmod{5}</math>, so that we can take <math>11^{13p + 1}</math> to the fifth root. Similarly, we want to look for the lowest power <math>n</math> of <math>7</math> such that <math>13n - 1 \equiv 0 \pmod{5}</math>. Again, this allows us to take the fifth root of <math>7^{13n - 1}</math>. Obviously, we want to add <math>1</math> to <math>13p</math> and subtract <math>1</math> from <math>13n</math> because <math>11^{13p}</math> and <math>7^{13n}</math> are multiplied by <math>11</math> and divided by <math>7</math>, respectively. With these conditions satisfied, we can simply multiply <math>11^{p}</math> and <math>7^{n}</math> and substitute this quantity into <math>y</math> to attain our answer. | Note that because <math>11</math> and <math>7</math> are prime, the minimum value of <math>x</math> must involve factors of <math>7</math> and <math>11</math> only. Thus, we try to look for the lowest power <math>p</math> of <math>11</math> such that <math>13p + 1 \equiv 0 \pmod{5}</math>, so that we can take <math>11^{13p + 1}</math> to the fifth root. Similarly, we want to look for the lowest power <math>n</math> of <math>7</math> such that <math>13n - 1 \equiv 0 \pmod{5}</math>. Again, this allows us to take the fifth root of <math>7^{13n - 1}</math>. Obviously, we want to add <math>1</math> to <math>13p</math> and subtract <math>1</math> from <math>13n</math> because <math>11^{13p}</math> and <math>7^{13n}</math> are multiplied by <math>11</math> and divided by <math>7</math>, respectively. With these conditions satisfied, we can simply multiply <math>11^{p}</math> and <math>7^{n}</math> and substitute this quantity into <math>y</math> to attain our answer. | ||
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We can simply look for suitable values for <math>p</math> and <math>n</math>. We find that the lowest <math>p</math>, in this case, would be <math>3</math> because <math>13(3) + 1 \equiv 0 \pmod{5}</math>. Moreover, the lowest <math>q</math> should be <math>2</math> because <math>13(2) - 1 \equiv 0 \pmod{5}</math>. Hence, we can substitute the quantity <math>11^{3} \cdot 7^{2}</math> into <math>y</math>. Doing so gets us: | We can simply look for suitable values for <math>p</math> and <math>n</math>. We find that the lowest <math>p</math>, in this case, would be <math>3</math> because <math>13(3) + 1 \equiv 0 \pmod{5}</math>. Moreover, the lowest <math>q</math> should be <math>2</math> because <math>13(2) - 1 \equiv 0 \pmod{5}</math>. Hence, we can substitute the quantity <math>11^{3} \cdot 7^{2}</math> into <math>y</math>. Doing so gets us: | ||
− | <cmath>(a^{5c}b^{5d}) = | + | <cmath>(a^{5c}b^{5d}) = \frac{11(11^{3} \cdot 7^{2})^{13}}{7} = 11^{40} \cdot 7^{25}.</cmath> |
Taking the fifth root of both sides, we are left with <math>a^cb^d = 11^{8} \cdot 7^{5}</math>. <math>a + b + c + d = 11 + 7 + 8 + 5 = \boxed{\textbf{(B)}\ 31}</math> | Taking the fifth root of both sides, we are left with <math>a^cb^d = 11^{8} \cdot 7^{5}</math>. <math>a + b + c + d = 11 + 7 + 8 + 5 = \boxed{\textbf{(B)}\ 31}</math> | ||
+ | ==Solution 2== | ||
+ | |||
+ | A simpler way to tackle this problem without all that modding is to keep the equation as: | ||
+ | |||
+ | <cmath>7*a^{5b}c^{5d} = 11y^{13}</cmath> | ||
+ | |||
+ | As stated above, <math>a</math> and <math>c</math> must be the factors 7 and 11 in order to keep <math>x</math> at a minimum. Moving all the non-y terms to the left hand side of the equation, we end up with: | ||
+ | |||
+ | <cmath>7^{5b+1}11^{5d-1}=y^{13}</cmath> | ||
+ | |||
+ | The above equation means that <math>y</math> must also contain only the factors 7 and 11 (again, in order to keep <math>x</math> at a minimum), so we end up with: | ||
+ | |||
+ | <cmath>7^{5b+1}11^{5d-1}=7^{13h}11^{13j}</cmath> | ||
+ | |||
+ | (<math>h</math> and <math>j</math> are arbitrary variables placed in order to show that <math>y</math> could have more than just one 7 or one 11 as factors) | ||
+ | |||
+ | Since 7 and 11 are prime, we know that <math>5b+1=13h</math> and <math>5d-1=13j</math>. The smallest positive combinations that would work are <math>b=5,h=2</math> and <math>d=8,j=3</math>. Therefore, <math>a+b+c+d=7+5+11+8=31</math>. <math>\boxed{\textbf{(B)}\ 31}</math> is correct. | ||
+ | |||
+ | ==Solution 3== | ||
+ | |||
+ | Another way to solve this problem solve for x. First, we can divide both sides by 7 to get: | ||
+ | |||
+ | <cmath>x^5 = \frac{11y^{13}}{7}</cmath> | ||
+ | |||
+ | Next, we take the fifth root on both sides, which gets us: | ||
+ | |||
+ | <cmath>x = \sqrt[5]{\frac{11y^{13}}{7}}</cmath> | ||
+ | |||
+ | <cmath>x = y^2 \cdot \sqrt[5]{\frac{11y^{3}}{7}}</cmath> | ||
+ | |||
+ | Since we know x is a positive integer that we are trying to minimize, we can let y equal the smallest number that will make x an integer. In this case, we let <math>y = 11^3 \cdot 7^2</math>(Make sure you see why this makes x the smallest integer possible!), which when plugged in, results in: | ||
+ | |||
+ | <cmath>x = (11^3 \cdot 7^2)^2 \cdot \sqrt[5]{\frac{11}{7}\cdot (11^3 \cdot 7^2)^3}</cmath> | ||
+ | |||
+ | <cmath>x = (11^3 \cdot 7^2)^2 \cdot \sqrt[5]{\frac{11}{7}\cdot 11^9 \cdot 7^6}</cmath> | ||
+ | |||
+ | <cmath>x = (11^3 \cdot 7^2)^2 \cdot \sqrt[5]{11^{10} \cdot 7^5}</cmath> | ||
+ | |||
+ | <cmath>x = (11^3 \cdot 7^2)^2 \cdot 11^2 \cdot 7^1</cmath> | ||
+ | |||
+ | <cmath>x = 7^5 \cdot 11^ 8</cmath> | ||
+ | |||
+ | This gets us <math>a^cb^d = 7^5\cdot 11^8</math>, so <math>a + b + c + d = 7 + 5 + 11 + 8 = \boxed{\textbf{(B)}\ 31}</math> | ||
+ | ~lucaswujc, <math>\LaTeX</math> help from Technodoggo | ||
+ | |||
+ | ==Solution 4 (easy)== | ||
+ | According to the problem, we have that <math>x^5</math> and <math>y^{13}</math> must be a multiple of both <math>7</math> and <math>11</math>. Thus, in their prime factorisation, there must be <math>7</math> and <math>11</math>. Thus, we have <math>a=7</math> and <math>b=11</math>. Now, let <math>x=7^c11^d\implies x^5=7^{5c}11^{5d}\implies7x^5=7^{5c+1}11^{5d}</math>. | ||
+ | We can now divide both sides by 11 in our original equation <math>7x^5=11y^{13}</math> to get <math>y^{13}=7^{5c+1}11^{5d-1}</math>. As we are only considering integers, we have <cmath>5c+1\equiv0~(\text{mod}~13)\implies5c\equiv12~(\text{mod}~13)</cmath> and <cmath>5d-1\equiv0~(\text{mod}~13)\implies5d\equiv1~(\text{mod}~13).</cmath> | ||
+ | |||
+ | We can apply brute force to solve for <math>c</math> and <math>d</math> as the numbers aren't big. For the first congruence, we find that <math>25</math> is the smallest number that satisfies, thus <math>c=5</math>. For the second congruence, we find that <math>40</math> is the smallest number that satisfies, thus <math>d=8</math>. Summarising, we get <math>a+b+c+d=7+11+5+8=\boxed{\textbf{(D)}~31}</math>. | ||
== See Also== | == See Also== | ||
{{AMC12 box|year=2003|ab=B|num-b=17|num-a=19}} | {{AMC12 box|year=2003|ab=B|num-b=17|num-a=19}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 10:07, 7 September 2023
Problem
Let and be positive integers such that The minimum possible value of has a prime factorization What is
Solution 1
Substitute into . We then have . Divide both sides by , and it follows that:
Note that because and are prime, the minimum value of must involve factors of and only. Thus, we try to look for the lowest power of such that , so that we can take to the fifth root. Similarly, we want to look for the lowest power of such that . Again, this allows us to take the fifth root of . Obviously, we want to add to and subtract from because and are multiplied by and divided by , respectively. With these conditions satisfied, we can simply multiply and and substitute this quantity into to attain our answer.
We can simply look for suitable values for and . We find that the lowest , in this case, would be because . Moreover, the lowest should be because . Hence, we can substitute the quantity into . Doing so gets us:
Taking the fifth root of both sides, we are left with .
Solution 2
A simpler way to tackle this problem without all that modding is to keep the equation as:
As stated above, and must be the factors 7 and 11 in order to keep at a minimum. Moving all the non-y terms to the left hand side of the equation, we end up with:
The above equation means that must also contain only the factors 7 and 11 (again, in order to keep at a minimum), so we end up with:
( and are arbitrary variables placed in order to show that could have more than just one 7 or one 11 as factors)
Since 7 and 11 are prime, we know that and . The smallest positive combinations that would work are and . Therefore, . is correct.
Solution 3
Another way to solve this problem solve for x. First, we can divide both sides by 7 to get:
Next, we take the fifth root on both sides, which gets us:
Since we know x is a positive integer that we are trying to minimize, we can let y equal the smallest number that will make x an integer. In this case, we let (Make sure you see why this makes x the smallest integer possible!), which when plugged in, results in:
This gets us , so ~lucaswujc, help from Technodoggo
Solution 4 (easy)
According to the problem, we have that and must be a multiple of both and . Thus, in their prime factorisation, there must be and . Thus, we have and . Now, let . We can now divide both sides by 11 in our original equation to get . As we are only considering integers, we have and
We can apply brute force to solve for and as the numbers aren't big. For the first congruence, we find that is the smallest number that satisfies, thus . For the second congruence, we find that is the smallest number that satisfies, thus . Summarising, we get .
See Also
2003 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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 |
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