Math Problem Statement

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

Solution

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:

    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:

    • 252^5: The exponent of 2 is 5, which is odd. We need to divide by 212^1 (i.e., divide by 2) to make it 242^4.
    • 323^2: The exponent of 3 is already even.
    • 515^1: The exponent of 5 is odd. We need to divide by 515^1 (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 21×51=2×5=102^1 \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.


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.

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Math Problem Analysis

Mathematical Concepts

Prime Factorization
Perfect Squares
Number Theory

Formulas

Prime factorization of a number
Conditions for a number to be a perfect square

Theorems

A number is a perfect square if all exponents in its prime factorization are even

Suitable Grade Level

Grades 6-8