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:
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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$$ -
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:
- : The exponent of 2 is 5, which is odd. We need to divide by (i.e., divide by 2) to make it .
- : The exponent of 3 is already even.
- : The exponent of 5 is odd. We need to divide by (i.e., divide by 5) to remove it.
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Divide by the product of the factors with odd exponents: To make 1440 a perfect square, we need to divide by .
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:
- What is the perfect square result after dividing 1440 by 10?
- How do we find the least number by which a non-square number should be multiplied to make it a perfect square?
- What are some other examples of making numbers into perfect squares by dividing?
- How does prime factorization help in identifying perfect squares?
- 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