$252$ can be expressed as a product of primes as

### Concept & Strategy Expressing a number "as a product of primes" simply means calculating its complete prime factorization. The most reliable method is creating a factor tree or using the ladder division method starting from the smallest prime. ### Step-by-Step Solution * **Step 1:** $252$ is even, so divide by $2$. $$252 \div 2 = 126$$ * **Step 2:** $126$ is also even, divide by $2$ again. $$126 \div 2 = 63$$ * **Step 3:** $63$ is not even. The sum of digits ($6+3=9$) is divisible by $3$. Divide by $3$. $$63 \div 3 = 21$$ * **Step 4:** $21$ is notoriously divisible by $3$. $$21 \div 3 = 7$$ * **Step 5:** $7$ is a prime number. Divide by $7$. $$7 \div 7 = 1$$ * **Calculation / Deduction:** Writing out all the divisors gathered in our steps: $252 = 2 \times 2 \times 3 \times 3 \times 7$ ### Exam Strategy & Shortcut **Direct Option Testing:** Since you are given the expressions, calculate their products quickly. * (a) $4 \times 9 \times 7 = 36 \times 7 = 252$. (Match found!) * (b) $8 \times 21 = 168$. (Incorrect) * (c) $81 \times 7 = 567$. (Incorrect) * (d) $54 \times 7 = 378$. (Incorrect) This method is often faster than performing the division steps yourself. ### Common Pitfall Dividing by composite numbers (like $4$ or $9$) initially to save time, and then forgetting to break those composite numbers down into primes at the final step. Always ensure every number in the final list is strictly prime. ### Final Answer **Therefore, the correct answer is $2 \times 2 \times 3 \times 3 \times 7$.**