To find the smallest natural number by which 1372 must be divided to obtain a perfect cube, we can follow these steps:
Step 1: Prime Factorization
First, we need to perform the prime factorization of 1372.
- Divide by 2: 1372 ÷ 2 = 686
- Divide by 2 again: 686 ÷ 2 = 343
- 343 is 7^3 (since 7 × 7 × 7 = 343)
Thus, we can express 1372 as:
- 1372 = 2^2 × 7^3
Step 2: Analyze the Exponents
For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.
- In our factorization:
- 2 has an exponent of 2
- 7 has an exponent of 3
Step 3: Determine the Necessary Adjustments
We need to adjust the exponent of 2 to the nearest multiple of 3.
- The nearest multiple of 3 greater than 2 is 3.
- To achieve this, we require one more factor of 2 (2^3 - 2^2 = 2^1).
Step 4: Calculate the Smallest Natural Number
To adjust the factorization to a perfect cube, we must divide by the additional factor:
- Therefore, we must divide by 2^1 = 2.
Conclusion
The smallest natural number by which 1372 must be divided to obtain a perfect cube is:
- Answer: 2