Difference between revisions of "2019 AIME II Problems/Problem 3"
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− | As 71 is prime, <math>c</math>, <math>d</math>, and <math>e</math> must be 1, 1, and 71 ( | + | 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>. |
Then the answer is simply <math>8 \times 12 = \boxed{096}</math>. | Then the answer is simply <math>8 \times 12 = \boxed{096}</math>. |
Revision as of 15:29, 23 June 2019
Problem 3
Find the number of -tuples of positive integers that satisfy the following systems of equations:
Solution
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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.