$\sqrt{11881} \times \sqrt{x} = 10137$

### Concept & Strategy Extract the large integer root first using estimation limits, isolate $\sqrt{x}$ via basic arithmetic division, and square the outcome. Approximation limits: $$ 100^2 = 10000 \quad \text{and} \quad 110^2 = 12100 $$ ### Step-by-Step Solution - **Step 1: Evaluate $\sqrt{11881}$.** - $11881$ lies perfectly between $10000$ ($100^2$) and $12100$ ($110^2$). - Since it ends in $1$, the unit digit of its root is either $1$ or $9$. - Given that $11881$ is much closer to $12100$, we check $109$. - Verification: $109^2 = (110 - 1)^2 = 12100 - 220 + 1 = 11881$. Thus, $\sqrt{11881} = 109$. - **Step 2: Isolate the unknown term.** - $109 \times \sqrt{x} = 10137$ - $\sqrt{x} = \frac{10137}{109}$ - Computing division: $\sqrt{x} = 93$ - **Step 3: Compute $x$.** - $x = 93^2$ - $93^2 = (100 - 7)^2 = 10000 - 1400 + 49 = 8649$ ### Exam Strategy & Shortcut Use last digit patterns for the division. In $109 \times \sqrt{x} = 10137$, the unit digit $9$ multiplied by some number must result in a unit digit of $7$. Since $9 \times 3 = 27$, $\sqrt{x}$ must end in $3$. Squaring a value ending in $3$ ($\dots3^2$) guarantees the final value $x$ must end in $9$. This filters options to $8649$ and $9409$. Knowing $90^2 = 8100$, $93^2$ must be close to $8100$, confirming $8649$. ### Common Pitfall Dividing $10137$ by $109$ incorrectly due to rushing. Take care with multi-digit divisions by rounding $109$ to $110$ mentally to check magnitude constraints. ### Final Answer Therefore, the correct answer is 8649.