Question

Given that $2^{2004}$ is a $604$ -digit number whose first digit is $1,$ how many elements of the set $S = \{2^0, 2^1, 2^2, \ldots, 2^{2003}\}$ have a first digit of $4?$

$194$

$195$

$196$

$197$

$198$

Solution:

The smallest power of $2$ with any given digit-count has leading digit $1.$ Since $2^{2004}$ has $604$ digits, there are $603$ elements of $S$ with leading digit $1.$

Whenever $2^k$ leads with $1,$ $2^{k+1}$ leads with $2$ or $3,$ and $2^{k+2}$ leads with $4, 5, 6,$ or $7.$ So $603$ elements lead with $2$ or $3,$ $603$ lead with $4$ through $7,$ and $2004 - 3(603) = 195$ lead with $8$ or $9.$

Finally, $2^k$ leads with $8$ or $9$ exactly when $2^{k-1}$ leads with $4,$ so there are $195$ elements with first digit $4.$

Thus, the correct answer is B.

2004 AMC 12B Problem 25

Solution:

Problem 25 in Other Years