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2005 AMC 12A Problems/Problem 21

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Problem

How many ordered triples of integers $(a,b,c)$ , with $a \ge 2$ , $b\ge 1$ , and $c \ge 0$ , satisfy both $\log_a b = c^{2005}$ and $a + b + c = 2005$ ?

$\mathrm{(A)} \ 0 \qquad \mathrm{(B)} \ 1 \qquad \mathrm{(C)} \ 2 \qquad \mathrm{(D)} \ 3 \qquad \mathrm{(E)} \ 4$

Solution

$a^{c^{2005}} = b$

Casework upon $c$ :

$c = 0$ : Then $a^0 = b \Longrightarrow b = 1$ . Thus we get $(2004,1,0)$ .
$c = 1$ : Then $a^1 = b \Longrightarrow a = b$ . Thus we get $(1002,1002,1)$ .
$c \ge 2$ : Then the exponent of $a$ becomes huge, and since $a \ge 2$ there is no way we can satisfy the second condition. Hence we have two ordered triples $\mathrm{(C)}$ .

See also

2005 AMC 12A (Problems • Resources)
Preceded by Problem 20	Followed by Problem 22
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

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/2005_AMC_12A_Problems/Problem_21"

Category: Introductory Algebra Problems

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