The least number by which 1440 should be divided in order to make it a perfect square number is dash

To find the least number by which 1440 should be divided to make it a perfect square, we need to follow these steps:

1. **Prime factorization of 1440:**
   \[
   1440 \div 2 = 720 \\
   720 \div 2 = 360 \\
   360 \div 2 = 180 \\
   180 \div 2 = 90 \\
   90 \div 2 = 45 \\
   45 \div 3 = 15 \\
   15 \div 3 = 5 \\
   5 \div 5 = 1
   \]
   So, the prime factorization of 1440 is:
   \[
   1440 = 2^5 \times 3^2 \times 5^1
   \]

2. **Make the powers of all primes even:**
   For a number to be a perfect square, all the exponents in its prime factorization must be even. From the factorization of 1440:
   - \( 2^5 \): The exponent of 2 is 5, which is odd. We need to divide by \( 2^1 \) (i.e., divide by 2) to make it \( 2^4 \).
   - \( 3^2 \): The exponent of 3 is already even.
   - \( 5^1 \): The exponent of 5 is odd. We need to divide by \( 5^1 \) (i.e., divide by 5) to remove it.

3. **Divide by the product of the factors with odd exponents:**
   To make 1440 a perfect square, we need to divide by \( 2^1 \times 5^1 = 2 \times 5 = 10 \).

Therefore, the least number by which 1440 should be divided to make it a perfect square is **10**.

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Do you want more details or have any questions?

Here are 5 related questions:
1. What is the perfect square result after dividing 1440 by 10?
2. How do we find the least number by which a non-square number should be multiplied to make it a perfect square?
3. What are some other examples of making numbers into perfect squares by dividing?
4. How does prime factorization help in identifying perfect squares?
5. What is the role of even powers in a perfect square?

**Tip:** Always check the prime factorization when working with perfect squares; all prime exponents must be even.

Learn how to find the least number by which 1440 should be divided to make it a perfect square using prime factorization. This problem demonstrates key number theory concepts and is suitable for students in Grades 6-8.

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