Sure, let's solve this problem step-by-step:
1. Understanding the problem: We need to determine the smallest number by which 1324 must be multiplied to make the product a perfect cube. A perfect cube is a number that can be expressed as [tex]\(x^3\)[/tex] where [tex]\(x\)[/tex] is an integer.
2. Prime factorization of 1324:
- We start by factorizing 1324 into its prime factors. The factorization goes as follows:
- [tex]\(1324 \div 2 = 662\)[/tex]
- [tex]\(662 \div 2 = 331\)[/tex]
- 331 is a prime number.
Therefore, the prime factors of 1324 are [tex]\(2^2 \times 331^1\)[/tex].
3. Analyzing the prime factors:
- A number is a perfect cube if all the exponents in its prime factorization are multiples of 3.
- For [tex]\(2^2\)[/tex], the exponent 2 is not a multiple of 3. Hence, we need one more factor of 2 to make it [tex]\(2^3\)[/tex].
- For [tex]\(331^1\)[/tex], the exponent 1 is not a multiple of 3. We need two more factors of 331 to make it [tex]\(331^3\)[/tex].
4. Combining the additional factors:
- To make [tex]\(2^2\)[/tex] a perfect cube, multiply by [tex]\(2^1 = 2\)[/tex].
- To make [tex]\(331^1\)[/tex] a perfect cube, multiply by [tex]\(331^2\)[/tex].
5. Calculating the smallest number required:
- Combining these, we need to multiply 1324 by [tex]\(2 \times 331^2\)[/tex].
6. Final computation:
- [tex]\(331^2 = 109561\)[/tex]
- Therefore, the smallest number we need to multiply 1324 by is [tex]\(2 \times 109561 = 219122\)[/tex].
Thus, the smallest number by which 1324 must be multiplied to make the product a perfect cube is 219122.
