Multiply $5746320819$ by $125$.

### Concept & Strategy Multiplying any enormous number by 125 is a prime candidate for the fraction substitution technique. Since $125 = 1000 / 8$, we can divide the large number by 8 and append three zeros. $$x \times 125 = x \times \frac{1000}{8}$$ ### Step-by-Step Solution * Rewrite 125 as a fraction of 1000: $$5746320819 \times \frac{1000}{8}$$ * Set up the division of the large number by 8: $$\frac{5746320819}{8} \times 1000$$ * Perform the step-by-step division: $57 \div 8 = 7$ (remainder 1) $14 \div 8 = 1$ (remainder 6) $66 \div 8 = 8$ (remainder 2) ... continuing this process yields 718290102. * Handle the decimal remainder for the final digits: The remainder at the end is 3. So we have 718290102 and 3/8. $3/8 \times 1000 = 375$. * Combine the whole and fractional parts: $$718290102375$$ ### Exam Strategy & Shortcut Look at the options before you divide! You only need to divide the first few digits to find the correct answer. $57 \div 8 = 7$ (Starts with 7). $14 \div 8 = 1$ (Second digit is 1). The prefix is 71. Look at the options: (a) 718,290... (b) 728,490... (c) 748,290... (d) 798,290... Option (a) is the only one that starts with 718. You can confidently select it after calculating just the first two digits! ### Common Pitfall Attempting a standard multi-line multiplication with 125. Even if you are fast at multiplying, adding up three offset rows of a 10-digit number virtually guarantees an addition or alignment error. ### Final Answer **Therefore, the correct answer is 718,290,102,375.**