A simple calculation will reveal that 72 has 12 factors including 1 and 72. How many natural numbers less than 100 have 12 such factors
To determine the count of natural numbers less than 100 that have 12 factors, we can use the concept of factorization.
To begin, we know that the number of factors of any positive integer can be determined using its prime factorization. So, let's find the numbers that have a prime factorization with exponents that multiply out to 12.
The number 72 has a prime factorization of 2^3 * 3^2, where the exponents are 3 and 2. To find the number of factors, we add 1 to each exponent and multiply the results. In this case, we would have (3 + 1) * (2 + 1) = 4 * 3 = 12 factors.
To generalize this, we need a number with two prime factors. We can list such numbers and calculate their factors:
1. 2^11 * 3^0 = 2048 → (11 + 1) * (0 + 1) = 12 factors
2. 2^0 * 3^11 = 177147 → (0 + 1) * (11 + 1) = 12 factors
3. 2^5 * 3^1 = 192 → (5 + 1) * (1 + 1) = 12 factors
4. 2^1 * 3^5 = 486 → (1 + 1) * (5 + 1) = 12 factors
So, there are 4 natural numbers less than 100 that have 12 factors.