Difference between revisions of "2019 AIME II Problems/Problem 3"
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− | ==Problem | + | ==Problem== |
Find the number of <math>7</math>-tuples of positive integers <math>(a,b,c,d,e,f,g)</math> that satisfy the following systems of equations: | Find the number of <math>7</math>-tuples of positive integers <math>(a,b,c,d,e,f,g)</math> that satisfy the following systems of equations: | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
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\end{align*}</cmath> | \end{align*}</cmath> | ||
− | ==Solution== | + | ==Solution 1== |
As 71 is prime, <math>c</math>, <math>d</math>, and <math>e</math> must be 1, 1, and 71 (in some order). However, since <math>c</math> and <math>e</math> are divisors of 70 and 72 respectively, the only possibility is <math>(c,d,e) = (1,71,1)</math>. Now we are left with finding the number of solutions <math>(a,b,f,g)</math> satisfying <math>ab = 70</math> and <math>fg = 72</math>, which separates easily into two subproblems. The number of positive integer solutions to <math>ab = 70</math> simply equals the number of divisors of 70 (as we can choose a divisor for <math>a</math>, which uniquely determines <math>b</math>). As <math>70 = 2^1 \cdot 5^1 \cdot 7^1</math>, we have <math>d(70) = (1+1)(1+1)(1+1) = 8</math> solutions. Similarly, <math>72 = 2^3 \cdot 3^2</math>, so <math>d(72) = 4 \times 3 = 12</math>. | As 71 is prime, <math>c</math>, <math>d</math>, and <math>e</math> must be 1, 1, and 71 (in some order). However, since <math>c</math> and <math>e</math> are divisors of 70 and 72 respectively, the only possibility is <math>(c,d,e) = (1,71,1)</math>. Now we are left with finding the number of solutions <math>(a,b,f,g)</math> satisfying <math>ab = 70</math> and <math>fg = 72</math>, which separates easily into two subproblems. The number of positive integer solutions to <math>ab = 70</math> simply equals the number of divisors of 70 (as we can choose a divisor for <math>a</math>, which uniquely determines <math>b</math>). As <math>70 = 2^1 \cdot 5^1 \cdot 7^1</math>, we have <math>d(70) = (1+1)(1+1)(1+1) = 8</math> solutions. Similarly, <math>72 = 2^3 \cdot 3^2</math>, so <math>d(72) = 4 \times 3 = 12</math>. | ||
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-scrabbler94 | -scrabbler94 | ||
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+ | ==Solution 2== | ||
+ | We know that any two consecutive numbers are coprime. Using this, we can figure out that <math>c=1</math> and <math>e=1</math>. <math>d</math> then has to be <math>71</math>. Now we have two equations left. <math>ab=70</math> and <math>fg=72</math>. To solve these we just need to figure out all of the factors. Doing the prime factorization of <math>70</math> and <math>72</math>, we find that they have <math>8</math> and <math>12</math> factors, respectively. The answer is <math>8 \times 12=\boxed{096}</math> | ||
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+ | ~Hithere22702 | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2019|n=II|num-b=2|num-a=4}} | {{AIME box|year=2019|n=II|num-b=2|num-a=4}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 02:29, 16 February 2021
Contents
Problem
Find the number of -tuples of positive integers that satisfy the following systems of equations:
Solution 1
As 71 is prime, , , and must be 1, 1, and 71 (in some order). However, since and are divisors of 70 and 72 respectively, the only possibility is . Now we are left with finding the number of solutions satisfying and , which separates easily into two subproblems. The number of positive integer solutions to simply equals the number of divisors of 70 (as we can choose a divisor for , which uniquely determines ). As , we have solutions. Similarly, , so .
Then the answer is simply .
-scrabbler94
Solution 2
We know that any two consecutive numbers are coprime. Using this, we can figure out that and . then has to be . Now we have two equations left. and . To solve these we just need to figure out all of the factors. Doing the prime factorization of and , we find that they have and factors, respectively. The answer is
~Hithere22702
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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