The least number by which $1470$ must be divided to get a number which is a perfect square, is

### Concept & Strategy To determine the smallest divisor needed to make a number a perfect square, find its prime factorization. Identify any prime factors that have odd exponents. Dividing by the product of those unpaired prime factors will remove them, leaving only even exponents (a perfect square). ### Step-by-Step Solution Find the prime factorization of $1470$. Extract the obvious factor of $10$: $1470 = 10 \times 147$ $10 = 2 \times 5$ Now factorize $147$. It is divisible by $3$ (since $1+4+7=12$): $147 = 3 \times 49$ Recognize $49$ as a perfect square: $49 = 7^2$ Combine all the prime factors: $1470 = 2^1 \times 3^1 \times 5^1 \times 7^2$ Check the exponents: * The power of $7$ is $2$ (Even). * The powers of $2$, $3$, and $5$ are all $1$ (Odd). To convert this expression into a perfect square via division, we must divide out the single unpaired factors of $2$, $3$, and $5$ so they cancel out completely. Required divisor $= 2 \times 3 \times 5 = 30$. ### Exam Strategy & Shortcut Spot the squares immediately. You know $1470 = 147 \times 10$. You should know $147 = 3 \times 49$. Since $49$ is already a square, set it aside. You are left with $3 \times 10 = 30$. To get rid of this non-square $30$, you must divide by exactly $30$. ### Common Pitfall Dividing out only one of the odd-powered prime factors instead of multiplying them all together to form the final divisor. All unpaired primes must be eliminated. ### Final Answer Therefore, the correct answer is **30**.