Problem of the Week
Problem D and Solution
Welcome to a New Year!

Problem

$5^3$ is a power with base $5$ and exponent $3.$

$5^3$ means $5\times 5\times 5$ and is equal to $125$ when expressed as an integer.

When $8^{674}\times 5^{2025}$ is expressed as an integer, how many digits are in the product?

Solution

An immediate temptation might be to reach for a calculator. In this case, basic calculator technology will let you down. We will solve this problem using our knowledge of powers and corresponding power laws. $$\begin{array}{rl}{8}^{674}\times 5^{2025}& =((2^3{)}^{674})\times 5^{2025}\\ & =2^{3\times 674}\times 5^{2025}\\ & =2^{2022}\times 5^{2025}\\ & =2^{2022}\times 5^{2022}\times 5^3\\ & =(2\times 5{)}^{2022}\times 125\\ & ={10}^{2022}\times 125\end{array}$$ But ${10}^{2022}$ is the number $1$ followed by $2022$ zeroes. When we multiply this number by the three-digit number $125$, we obtain the number $125$ followed by $2022$ zeroes. Therefore, $8^{674}\times 5^{2025}$ has $2022+3=2025$ digits. Happy New Year again!