$935421 \times 625 = x$

### Concept & Formula Multiplying a large number by 625 can be vastly simplified by expressing 625 as a power of 10 divided by a power of 2. $$625 = \frac{10000}{16}$$ ### Step-by-Step Solution * Rewrite the mathematical expression by replacing 625 with its fractional equivalent. $$935421 \times \frac{10000}{16}$$ * This means we must divide $9354210000$ by 16. * First digits: $935 \div 16 = 58$ with a remainder of 7. * Carry down the 4 to get 74. $74 \div 16 = 4$ with a remainder of 10. * Continue this division sequence: $9354210000 \div 16 = 584638125$. ### Exam Strategy & Shortcut Use the **Digital Root (Digit Sum)** method for immediate elimination. Sum of digits of $935421 = 9+3+5+4+2+1 = 24 \rightarrow 6$. Sum of digits of $625 = 6+2+5 = 13 \rightarrow 4$. Product digital root = $6 \times 4 = 24 \rightarrow 6$. Check the digital roots of the options: (a) 5+7+5+6+4+8+1+2+5 $\rightarrow$ 7 (b) 5+8+4+6+3+8+1+2+5 $\rightarrow$ 6 (Matches) (c) 5+8+4+6+4+9+1+2+5 $\rightarrow$ 8 (d) 5+8+5+6+2+8+1+2+5 $\rightarrow$ 6 Options (b) and (d) remain. Use basic estimation: $935 \times 625$ will be slightly more than $900 \times 600 = 540000$. Thus, $584$ is the correct starting sequence over $585$. ### Common Pitfall Attempting brute-force multiplication without converting 625 to a fraction. This consumes massive amounts of time and increases the risk of simple arithmetic errors in an exam setting. ### Final Answer Therefore, the correct answer is **584638125**.