what is the sum of the two least natural numbers that each have exactly six distinct,positive factors?
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To find the sum of the two least natural numbers that each have exactly six distinct, positive factors, we need to understand how to determine the number of factors of a given number.
First, we need to find the prime factorization of the number. A prime factorization represents a number as a product of prime numbers. For example, the prime factorization of 24 is 2 × 2 × 2 × 3, as 2 and 3 are both prime numbers.
To find the number of factors, we need to consider the exponents of the prime factors in the prime factorization. We increment each exponent by one and then multiply them together. For instance, using the prime factorization of 24, we have (2+1) × (2+1) × (1+1) = 3 × 3 × 2 = 18 factors.
Next, we can determine the factors by listing out all the combinations of the prime factors. In the case of 24, the factors are 1, 2, 3, 4, 6, 8, 12, and 24.
Now, let's find the two least natural numbers with exactly six distinct, positive factors.
Starting with the prime factorization, we know that both numbers must have only one prime factor raised to the power of 5, as 2 × 2 × 2 × 3 × 3 = 72, which has more than six factors.
We can then consider the two smallest prime numbers, 2 and 3. The next steps involve calculating the powers of each prime number:
Case 1: One prime factor is 2 and the other is 3.
- 2^5 × 3^1 = 96 (number of factors = 6)
- Factors: 1, 2, 3, 4, 6, 12
Case 2: One prime factor is 2 and the other is 3.
- 2^1 × 3^5 = 486 (number of factors = 6)
- Factors: 1, 2, 3, 6, 9, 18
The two least natural numbers that each have exactly six distinct, positive factors are 96 and 486. To find their sum, we can simply add them: 96 + 486 = 582.
Therefore, the sum of the two least natural numbers with exactly six distinct, positive factors is 582.