If $1537*$ is a perfect square, then the digit which replaces $*$ is

### Concept & Strategy To find a missing digit in a perfect square, estimate the root by finding the nearest known perfect squares, then square numbers in that specific narrow range to find the exact match. ### Step-by-Step Solution The number falls in the $15300$ range. Let's establish a base estimate. We know $100^2 = 10000$ and $120^2 = 14400$. The target number $1537*$ is larger than $14400$, so the square root must be greater than $120$. Let's test $125^2$. $125^2 = 15625$. The target $1537*$ is smaller than $15625$. Thus, the root must lie precisely between $120$ and $125$. Test the integers in this narrow window: $123^2 = 15129$ (Too small) $124^2 = 15376$ (This matches the $1537$ prefix perfectly!) Therefore, the complete number is $15376$, which means the missing final digit is $6$. ### Exam Strategy & Shortcut Look at the options: 2, 4, 5, 6. A perfect square can NEVER end in 2, 3, 7, or 8. This instantly eliminates option (a). If a perfect square ends in 5, the preceding digit MUST be 2 (since squares ending in 5 always end in 25). Here, the preceding digit is 7, so option (c) is eliminated. We are left with 4 and 6. Check $124^2$ since $120^2$ is $14400$. $124 \times 124 = 15376$. ### Common Pitfall Blindly testing all squares from $100$ upwards or trying to extract a square root of an incomplete number using long division. This wastes precious time. Always use bounding squares to narrow the search radius immediately. ### Final Answer Therefore, the correct answer is **6**.