- Find a number that meets the following conditions. – The value of the number is greater than 900 but less than 1000. – One of its prime factors has an exponent of 2. – When expressed in exponential form, its prime factorization has 4 bases.
The Correct Answer and Explanation is:
To find the number that satisfies all the given conditions, let’s break them down:
Conditions:
- The number is greater than 900 but less than 1000.
- The number should lie in the range 900 < N < 1000.
- One of its prime factors has an exponent of 2.
- This means the prime factorization of the number includes a prime raised to the power of 2, e.g., p2p^2p2, where ppp is a prime number.
- When expressed in exponential form, its prime factorization has 4 bases.
- The number should have exactly four different prime factors.
Step-by-Step Approach:
To find a number that fits all these conditions, we need to follow these steps:
1. Identify a number between 900 and 1000.
Start by looking for numbers between 900 and 1000 that have a prime factorization with 4 distinct prime factors.
2. Prime factorization with an exponent of 2.
The number must have one of its prime factors raised to the power of 2. So, we should look for numbers where one of the primes appears twice in the factorization.
3. Check for a prime factorization with exactly 4 distinct primes.
We need a number that has exactly four different prime factors.
Example:
Let’s check the number 960.
- Is 960 between 900 and 1000?
Yes, 960 is between 900 and 1000. - Prime factorization of 960: 960=26×3×5960 = 2^6 \times 3 \times 5960=26×3×5
- This factorization includes 3 different primes: 2,3,2, 3,2,3, and 555, but we need four distinct primes.
Let’s try another number, 990.
- Is 990 between 900 and 1000?
Yes, 990 is between 900 and 1000. - Prime factorization of 990: 990=2×32×5×11990 = 2 \times 3^2 \times 5 \times 11990=2×32×5×11
- This factorization includes four distinct primes: 2,3,5,2, 3, 5,2,3,5, and 111111, and one of them, 333, is raised to the power of 2.
Thus, 990 satisfies all the conditions:
- It’s between 900 and 1000.
- One of its prime factors, 323^232, has an exponent of 2.
- Its prime factorization includes four distinct primes: 2,3,5,112, 3, 5, 112,3,5,11.
Final Answer: 990
