To find the least number by which 1440 should be divided to get a perfect square, we need to analyze its prime factorization.

Step 1: Prime Factorization of 1440

Let's start by factoring 1440 into its prime factors.

$1440 \div 2 = 720$ $720 \div 2 = 360$ $360 \div 2 = 180$ $180 \div 2 = 90$ $90 \div 2 = 45$ $45 \div 3 = 15$ $15 \div 3 = 5$ $5 \div 5 = 1$

So, the prime factorization of 1440 is: $1440 = 2^5 \times 3^2 \times 5$

Step 2: Identify Which Prime Factors Need Adjustment

For a number to be a perfect square, all the powers of the prime factors must be even. In the prime factorization of 1440:

• $2^5$ has an odd power (5), so it needs one more factor of 2 to make it even.
• $3^2$ has an even power (2), so it's already perfect.
• $5^1$ has an odd power (1), so it needs one more factor of 5 to make it even.

Step 3: Find the Least Number

To make the powers of 2 and 5 even, we need to divide by $2 \times 5 = 10$.

Conclusion:

The least number by which 1440 should be divided to make it a perfect square is 10.

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Would you like further details or have any questions?

Here are 5 related questions you might find interesting:

1. How can you find the least number by which a given number should be multiplied to get a perfect square?
2. What is the perfect square that results from dividing 1440 by 10?
3. How do prime factorizations help in solving perfect square problems?
4. How can you use prime factorization to find the greatest common divisor (GCD)?
5. What is the smallest perfect square divisible by 1440?

Tip: When working with prime factorizations, always check for even powers of primes to identify perfect squares.