2023 AMC 10A Problems/Problem 5

Problem

How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$ ?

$\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad$

Solution 1

Prime factorizing this gives us $2^{15}\cdot3^{5}\cdot5^{15}$ Pairing $2^{15}$ and $5^{15}$ gives us a number with $15$ zeros giving us 15 digits. $3^5=243$ and this adds an extra 3 digits. $15+3=\text{\boxed{(E)18}}$

Page

Toolbox

Search

2023 AMC 10A Problems/Problem 5

Problem

Solution 1