Find the smallest number by which 576 should be divided to get a perfect cube.
The correct answer and explanation is:
To find the smallest number by which 576 should be divided to get a perfect cube, let’s follow these steps:
Step 1: Prime Factorization of 576
Start by factoring 576 into prime numbers. We can do this through repeated division.
- 576 ÷ 2 = 288
- 288 ÷ 2 = 144
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
Thus, the prime factorization of 576 is:
576=26×32576 = 2^6 \times 3^2
Step 2: Identify Perfect Cube Conditions
For a number to be a perfect cube, all the powers of its prime factors must be multiples of 3. Looking at the prime factorization:
- 262^6 is not a perfect cube because 6 is not a multiple of 3.
- 323^2 is not a perfect cube because 2 is not a multiple of 3.
Step 3: Make the Powers of Prime Factors Multiples of 3
- For 262^6, the closest multiple of 3 greater than or equal to 6 is 6 itself, so no adjustment is needed here.
- For 323^2, we need to multiply by 313^1 to make the power 3.
Thus, to make 576 a perfect cube, we must multiply by 313^1. However, we want the smallest number by which 576 should be divided. So, we need to divide by 323^2 to remove the excess power of 3. Dividing by 32=93^2 = 9 makes the number a perfect cube.
Final Answer:
The smallest number by which 576 should be divided to get a perfect cube is 9.
Explanation:
The number 576 has prime factors 262^6 and 323^2. To form a perfect cube, we need to adjust the powers of the prime factors so that each is a multiple of 3. By dividing 576 by 323^2, the power of 3 becomes 0, and the power of 2 is already a multiple of 3. Therefore, the smallest number to divide 576 by to get a perfect cube is 9.