In an algebraic simplification problem, evaluate the expression obtained when 111 a^2 b^2 c^2 is divided by 37 a^2, and express the result in its simplest form.

Introduction / Context:
This question checks your ability to simplify algebraic expressions that involve coefficients and powers of variables. The expression 111 a^2 b^2 c^2 divided by 37 a^2 requires you to handle both numerical division and cancellation of powers of a common variable. Such simplification problems are common in basic algebra and aptitude tests, where quick manipulation is important.

Given Data / Assumptions:

• The original expression is 111 a^2 b^2 c^2.
• This expression is divided by 37 a^2.
• a, b, and c are assumed to be nonzero real numbers so that division by a^2 is valid.
• We must simplify the resulting expression completely.

Concept / Approach:
The key ideas are to divide coefficients (numerical parts) and to subtract exponents of the same base when dividing powers. For the coefficient, 111 divided by 37 should be simplified. For the variables, a^2 in the numerator divided by a^2 in the denominator results in a^0, which equals 1. The powers of b and c have no counterparts in the denominator and so remain unchanged. Combining these observations produces a very simple final expression.

Step-by-Step Solution:
Step 1: Write the division explicitly as a fraction: (111 a^2 b^2 c^2) / (37 a^2).Step 2: Separate the numerical part and the variable part: (111/37) × (a^2/a^2) × b^2 c^2.Step 3: Simplify the numerical coefficient: 111/37 = 3, because 111 = 3 × 37.Step 4: Simplify the powers of a using the rule a^m / a^n = a^(m − n): a^2/a^2 = a^0 = 1.Step 5: The remaining variables are b^2 and c^2, which are unchanged.Step 6: Therefore the expression simplifies to 3 × 1 × b^2 c^2 = 3b^2c^2.

Verification / Alternative check:
To double check, substitute simple nonzero values for the variables. For example, let a = 1, b = 2, and c = 3. The original numerator is 111 × 1^2 × 2^2 × 3^2 = 111 × 4 × 9 = 3996. The denominator is 37 × 1^2 = 37. Then 3996/37 = 108. Now compute 3b^2c^2 with the same values: 3 × 4 × 9 = 108. Since both methods give 108, the simplification is correct.

Why Other Options Are Wrong:
The expressions 2c^2 and 2b^2 ignore proper simplification of the coefficient and drop one of the variable factors. The option 3 omits b^2 and c^2 entirely and is incomplete. The option 3bc^2 keeps only one power of b and thus fails to match the original exponent b^2. Only 3b^2c^2 retains the correct powers of b and c while using the proper numerical coefficient after division by 37 a^2.

Common Pitfalls:
Students sometimes forget that powers subtract when dividing or mistakenly add exponents instead. Another frequent mistake is not simplifying the fraction 111/37 or miscalculating it. Carefully separating coefficient simplification from variable exponent rules prevents these errors. Always check that the final expression preserves the correct powers for each variable from the original term.

Final Answer:
The simplified result of dividing 111 a^2 b^2 c^2 by 37 a^2 is 3b^2c^2.