What is the smallest number by which $3600$ be divided to make it a perfect cube?

Concept & Formula For a number to be a perfect cube, the exponent of each of its prime factors must be a multiple of $3$. When asked what to **divide** by, we must remove the prime factors whose exponents are not multiples of $3$, or reduce their exponents down to the nearest multiple of $3$ (including $0$, meaning the prime factor is completely removed). Step-by-Step Solution * **Given number**: $3600$ * Break down $3600$ into its prime factors. $$3600 = 36 \times 100$$ $$3600 = (6^2) \times (10^2)$$ $$3600 = (2 \times 3)^2 \times (2 \times 5)^2$$ $$3600 = 2^2 \times 3^2 \times 2^2 \times 5^2$$ * Combine the common bases: $$3600 = 2^4 \times 3^2 \times 5^2$$ * Now, analyze the exponents to find what must be removed (divided out) to leave only multiples of $3$: * For base $2$: Current power is $4$. The nearest multiple of $3$ below it is $3$. We must divide by $2^1$ to leave $2^3$. * For base $3$: Current power is $2$. The nearest multiple of $3$ below it is $0$. We must divide by $3^2$ to remove it. * For base $5$: Current power is $2$. The nearest multiple of $3$ below it is $0$. We must divide by $5^2$ to remove it. * The required divisor is the product of the removed parts: $$\text{Divisor} = 2^1 \times 3^2 \times 5^2 = 2 \times 9 \times 25$$ $$\text{Divisor} = 18 \times 25 = 450$$ Exam Strategy & Shortcut Instead of heavy factorization, use the multiple-choice options to your advantage by testing them directly on $3600$: * Option (a): $3600 \div 9 = 400$ (Not a cube, $20^2$) * Option (b): $3600 \div 50 = 72$ (Not a cube) * Option (c): $3600 \div 300 = 12$ (Not a cube) * Option (d): $3600 \div 450 = 8$ (Perfect cube! $8 = 2^3$) Testing the options backward takes seconds and guarantees accuracy. Common Pitfall Confusing "divided by" with "multiplied by". If the question asked what to *multiply* by, we would add factors to reach the next multiple of $3$ (requiring $2^2 \times 3^1 \times 5^1 = 60$). Make sure you read the operative word carefully. Final Answer **Therefore, the correct answer is 450.**