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Determining the Least Integer to Multiply to 4394 to Obtain a Perfect Cube
Determining the Least Integer to Multiply to 4394 to Obtain a Perfect Cube
In mathematics, the concept of a perfect cube is quite intriguing. A perfect cube is a number that is the cube of an integer. For instance, 125 is a perfect cube since 53 125. However, the query, “What is the least number that must be multiplied by 4394 so that it will become a perfect cube,” requires a more nuanced understanding of integers and their properties.
Understanding the Problem
Initially, one might think that finding the least integer to multiply by 4394 for it to become a perfect cube would be straightforward. However, the solution lies in the prime factorization of 4394 and the identified gaps in its cube factors.
Prime Factorization
To solve this, let’s first consider the prime factorization of 4394. Breaking down 4394, we find:
4394 2 × 2197
Noting that 2197 is a perfect cube (since (2197 13^3)), we then analyze the prime factorization of 2197:
2197 (13^3)
Consequently, 4394 2 × (13^3).
Identifying the Missing Factors
Lets break down the multiplication process to find the least integer needed. For a number to be a perfect cube, all prime factors must have powers that are multiples of 3. Here, 4394 is missing one 2 to become (2^3) and can be made a perfect cube by ensuring all prime factors are cubes.
Finding the Least Integer
Given that 4394 2 × (13^3), we need to multiply 4394 by 2 to balance the power of 2 to 3, thus:
4394 × 2 2 × (13^3) × 2 (2^3) × (13^3)
The cube root of (2^3) × (13^3) is:
Cube root of (2^3) × (13^3) 2 × 13 26
Hence, the least integer that must be multiplied to 4394 to obtain a perfect cube is 2.
Mathematical Insight
For any number that is double a perfect cube, the least number to be multiplied to make it a perfect cube is 4. This is because:
(n^3 cdot 2 cdot 4 n^3 cdot 2^3)
The cube root of (n^3 cdot 2^3) is (n cdot 2). Here, n 13, thus the least number is 2.
Conclusion
In conclusion, the least integer that must be multiplied by 4394 to make it a perfect cube is 2. This solution underscores the importance of prime factorization in number theory and the properties of perfect cubes. This approach not only provides a clear answer but also lays a solid foundational understanding of integer multiplication, prime factors, and perfect cubes.