Let $x$ be the product of two numbers $3,659,893,456,789,325,678$ and $342,973,489,379,256$. The number of digits in $x$ is

### Concept & Strategy The number of digits in the product of two large numbers can be determined by representing them in standard scientific notation (powers of 10). If number $A$ has $m$ digits and $B$ has $n$ digits, their product $A \times B$ will have either $(m+n-1)$ or $(m+n)$ digits, depending on whether the product of their leading digits exceeds $10$. ### Step-by-Step Solution * Count the digits in the first number: $A = 3,659,893,456,789,325,678$. This is a $19$-digit number. It can be approximated as $3.66 \times 10^{18}$. * Count the digits in the second number: $B = 342,973,489,379,256$. This is a $15$-digit number. It can be approximated as $3.43 \times 10^{14}$. * Find the product $x = A \times B$: $$x \approx (3.66 \times 10^{18}) \times (3.43 \times 10^{14})$$ $$x \approx (3.66 \times 3.43) \times 10^{32}$$ * Calculate the approximate product of the leading coefficients: $3.66 \times 3.43 \approx 12.55$ * So, $x \approx 12.55 \times 10^{32} = 1.255 \times 10^{33}$. * A number written as $C \times 10^k$ (where $1 \le C < 10$) has $k + 1$ digits. * Number of digits = $33 + 1 = 34$. ### Exam Strategy & Shortcut Use the digit sum rule: Number of digits = $m + n$ (if the product of the first digits of each number $\ge 10$) or $m + n - 1$ (if $< 10$). Here, $m = 19$ and $n = 15$. The leading digits are $3.6\dots$ and $3.4\dots$. Their product is $3.6 \times 3.4 > 10$. Therefore, digits = $19 + 15 = 34$. ### Common Pitfall Students often simply add the number of digits $(19 + 15 = 34)$ without checking the product of the leading digits. While it works here, if the numbers started with $1$ and $2$ (e.g., $1.5 \times 2.5 = 3.75$), the answer would be $19 + 15 - 1 = 33$. Always verify the carry-over. ### Final Answer Therefore, the correct answer is **34**.