2005 AMC 12B Problem 17

Question

How many distinct four-tuples

(a, b, c, d)

of rational numbers are there with

\begin{aligned} &a\log_{10} 2 + b\log_{10} 3 \\ &\quad {}+ c\log_{10} 5 + d\log_{10} 7 = 2005? \end{aligned}

0

1

17

2004

infinitely many

Rewrite the left side as

\log_{10}\left(2^a 3^b 5^c 7^d\right)

Then

2^a 3^b 5^c 7^d = 10^{2005} = 2^{2005}5^{2005}

; match exponents via unique factorization

The equation is equivalent to

\log_{10}\left(2^a 3^b 5^c 7^d\right) = 2005,

so

2^a 3^b 5^c 7^d = 10^{2005} = 2^{2005} \cdot 5^{2005}.

Clearing the denominators of

a, b, c, d

with a common integer multiplier and using the uniqueness of prime factorization, the exponents must match:

a = 2005,

b = 0,

c = 2005,

and

d = 0.

So there is exactly

1

such four-tuple.

Thus, the correct answer is B.

Problem 17 in other years