Difference between revisions of "2007 AMC 12B Problems/Problem 24"
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− | ==Problem | + | == Problem == |
How many pairs of positive integers <math>(a,b)</math> are there such that <math>\text{gcd}(a,b)=1</math> and <math>\frac{a}{b} + \frac{14b}{9a}</math> is an integer? | How many pairs of positive integers <math>(a,b)</math> are there such that <math>\text{gcd}(a,b)=1</math> and <math>\frac{a}{b} + \frac{14b}{9a}</math> is an integer? | ||
− | <math>\mathrm {(A)} 4\quad\mathrm {(B)} 6\quad\mathrm {(C)} 9\quad\mathrm {(D)} 12\quad\mathrm {(E)} \text{infinitely many}</math> | + | <math>\mathrm {(A)}\ 4\quad\mathrm {(B)}\ 6\quad\mathrm {(C)}\ 9\quad\mathrm {(D)}\ 12\quad\mathrm {(E)}\ \text{infinitely many}</math> |
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+ | == Solutions == | ||
+ | === Solution 1 === | ||
Combining the fraction, <math>\frac{9a^2 + 14b^2}{9ab}</math> must be an integer. | Combining the fraction, <math>\frac{9a^2 + 14b^2}{9ab}</math> must be an integer. | ||
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Thus there are four solutions: <math>(1,3)</math>, <math>(2,3)</math>, <math>(7,3)</math>, <math>(14,3)</math> and the answer is <math>\mathrm{(A)}</math> | Thus there are four solutions: <math>(1,3)</math>, <math>(2,3)</math>, <math>(7,3)</math>, <math>(14,3)</math> and the answer is <math>\mathrm{(A)}</math> | ||
− | ==Solution 2== | + | === Solution 2 === |
Let's assume that <math>\frac{a}{b} + \frac{14b}{9a} = m</math> We get | Let's assume that <math>\frac{a}{b} + \frac{14b}{9a} = m</math> We get | ||
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Similar reason for the fourth equation. This eliminates the last four solutions out of the above eight listed, giving us 4 solutions total <math>\mathrm {(A)}</math> | Similar reason for the fourth equation. This eliminates the last four solutions out of the above eight listed, giving us 4 solutions total <math>\mathrm {(A)}</math> | ||
− | ==Solution 3== | + | === Solution 3 === |
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Let <math>u = \frac{a}{b}</math>. Then the given equation becomes <math>u + \frac{14}{9u} = \frac{9u^2 + 14}{9u}</math>. | Let <math>u = \frac{a}{b}</math>. Then the given equation becomes <math>u + \frac{14}{9u} = \frac{9u^2 + 14}{9u}</math>. | ||
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Looking back at our equations for <math>m</math> and <math>n</math>, we can solve for <math>k = \frac{2m + 2n}{18} = \frac{m+n}{9}</math>. Since <math>k</math> is an integer, there are only <math>2</math> pairs of <math>(m,n)</math> that work: <math>(3,42)</math> and <math>(6,21)</math>. This means that there are <math>2</math> values of <math>k</math> such that <math>u</math> is an integer. But looking back at <math>u</math> in terms of <math>k</math>, we have <math>\pm</math>, meaning that there are <math>2</math> values of <math>u</math> for every <math>k</math>. Thus, the answer is <math>2 \cdot 2 = 4 \Rightarrow \mathrm{(A)}</math>. | Looking back at our equations for <math>m</math> and <math>n</math>, we can solve for <math>k = \frac{2m + 2n}{18} = \frac{m+n}{9}</math>. Since <math>k</math> is an integer, there are only <math>2</math> pairs of <math>(m,n)</math> that work: <math>(3,42)</math> and <math>(6,21)</math>. This means that there are <math>2</math> values of <math>k</math> such that <math>u</math> is an integer. But looking back at <math>u</math> in terms of <math>k</math>, we have <math>\pm</math>, meaning that there are <math>2</math> values of <math>u</math> for every <math>k</math>. Thus, the answer is <math>2 \cdot 2 = 4 \Rightarrow \mathrm{(A)}</math>. | ||
− | ==Solution 4== | + | === Solution 4 === |
− | |||
Rewriting the expression over a common denominator yields <math>\frac{9a^2 + 14b^2}{9ab}</math>. This expression must be equal to some integer <math>m</math>. | Rewriting the expression over a common denominator yields <math>\frac{9a^2 + 14b^2}{9ab}</math>. This expression must be equal to some integer <math>m</math>. | ||
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Thus, <math>\frac{9a^2 + 14b^2}{9ab} = m \rightarrow 9a^2 + 14b^2 = 9abm</math>. Taking this <math>\pmod{a}</math> yields <math>14b^2 \equiv 0\pmod{a}</math>. Since <math>\gcd(a,b)=1</math>, <math>14 \equiv 0\pmod{a}</math>. This implies that <math>a|14</math> so <math>a = 1, 2, 7, 14</math>. | Thus, <math>\frac{9a^2 + 14b^2}{9ab} = m \rightarrow 9a^2 + 14b^2 = 9abm</math>. Taking this <math>\pmod{a}</math> yields <math>14b^2 \equiv 0\pmod{a}</math>. Since <math>\gcd(a,b)=1</math>, <math>14 \equiv 0\pmod{a}</math>. This implies that <math>a|14</math> so <math>a = 1, 2, 7, 14</math>. | ||
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We can then take <math>9a^2 + 14b^2 = 9abm \pmod{b}</math> to get that <math>9 \equiv 0 \pmod{b}</math>. Thus <math>b = 1, 3, 9</math>. | We can then take <math>9a^2 + 14b^2 = 9abm \pmod{b}</math> to get that <math>9 \equiv 0 \pmod{b}</math>. Thus <math>b = 1, 3, 9</math>. | ||
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However, taking <math>9a^2 + 14b^2 = 9abm \pmod{3}</math>, <math>b^2 \equiv 0\pmod{3}</math> so <math>b</math> cannot equal 1. | However, taking <math>9a^2 + 14b^2 = 9abm \pmod{3}</math>, <math>b^2 \equiv 0\pmod{3}</math> so <math>b</math> cannot equal 1. | ||
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Also, note that if <math>b = 9</math>, <math>\frac{a}{b}+\frac{14b}{9a} = \frac{a}{9}+\frac{14}{a}</math>. Since <math>a|14</math>, <math>\frac{14}{a}</math> will be an integer, but <math>\frac{a}{9}</math> will not be an integer since none of the possible values of <math>a</math> are multiples of 9. Thus, <math>b</math> cannot equal 9. | Also, note that if <math>b = 9</math>, <math>\frac{a}{b}+\frac{14b}{9a} = \frac{a}{9}+\frac{14}{a}</math>. Since <math>a|14</math>, <math>\frac{14}{a}</math> will be an integer, but <math>\frac{a}{9}</math> will not be an integer since none of the possible values of <math>a</math> are multiples of 9. Thus, <math>b</math> cannot equal 9. | ||
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Thus, the only possible values of <math>b</math> is 3, and <math>a</math> can be 1, 2, 7, or 14. This yields 4 possible solutions, so the answer is <math>\mathrm{(A)}</math>. | Thus, the only possible values of <math>b</math> is 3, and <math>a</math> can be 1, 2, 7, or 14. This yields 4 possible solutions, so the answer is <math>\mathrm{(A)}</math>. | ||
− | ==Solution 5 (Similar to Solution 1)== | + | === Solution 5 (Similar to Solution 1) === |
− | |||
Rewriting <math>\frac{a}{b} + \frac{14b}{9a}</math> over a common denominator gives <math>\frac{9a^2 + 14b^2}{9ab}.</math> | Rewriting <math>\frac{a}{b} + \frac{14b}{9a}</math> over a common denominator gives <math>\frac{9a^2 + 14b^2}{9ab}.</math> | ||
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~coolmath2017 | ~coolmath2017 | ||
− | ==See Also== | + | == See Also == |
{{AMC12 box|year=2007|ab=B|num-b=23|num-a=25}} | {{AMC12 box|year=2007|ab=B|num-b=23|num-a=25}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 01:05, 19 October 2020
Contents
Problem
How many pairs of positive integers are there such that and is an integer?
Solutions
Solution 1
Combining the fraction, must be an integer.
Since the denominator contains a factor of ,
Since for some positive integer , we can rewrite the fraction(divide by on both top and bottom) as
Since the denominator now contains a factor of , we get .
But since , we must have , and thus .
For the original fraction simplifies to .
For that to be an integer, must be a factor of , and therefore we must have . Each of these values does indeed yield an integer.
Thus there are four solutions: , , , and the answer is
Solution 2
Let's assume that We get
Factoring this, we get equations-
(It's all negative, because if we had positive signs, would be the opposite sign of )
Now we look at these, and see that-
This gives us solutions, but we note that the middle term needs to give you back .
For example, in the case
, the middle term is , which is not equal by for any integer .
Similar reason for the fourth equation. This eliminates the last four solutions out of the above eight listed, giving us 4 solutions total
Solution 3
Let . Then the given equation becomes .
Let's set this equal to some value, .
Clearing the denominator and simplifying, we get a quadratic in terms of :
Since and are integers, is a rational number. This means that is an integer.
Let . Squaring and rearranging yields:
.
In order for both and to be an integer, and must both be odd or even. (This is easily proven using modular arithmetic.) In the case of this problem, both must be even. Let and .
Then:
.
Factoring 126, we get pairs of numbers: and .
Looking back at our equations for and , we can solve for . Since is an integer, there are only pairs of that work: and . This means that there are values of such that is an integer. But looking back at in terms of , we have , meaning that there are values of for every . Thus, the answer is .
Solution 4
Rewriting the expression over a common denominator yields . This expression must be equal to some integer .
Thus, . Taking this yields . Since , . This implies that so .
We can then take to get that . Thus .
However, taking , so cannot equal 1.
Also, note that if , . Since , will be an integer, but will not be an integer since none of the possible values of are multiples of 9. Thus, cannot equal 9.
Thus, the only possible values of is 3, and can be 1, 2, 7, or 14. This yields 4 possible solutions, so the answer is .
Solution 5 (Similar to Solution 1)
Rewriting over a common denominator gives
Thus, we have
Next, we have
Thus,
Next, we have
Thus,
Now, we simply do casework on
Plugging in and gives that there are total solutions for
~coolmath2017
See Also
2007 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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.