Final answer:
To find the number of positive integers that are divisors of at least one of $12^{12}$, $10^{10}$, and $15^{15}$, we need to find the prime factorizations of these numbers and count the unique divisors. The prime factorization of $12^{12}$ is $2^{24} \times 3^{12}$, the prime factorization of $10^{10}$ is $2^{10} \times 5^{10}$, and the prime factorization of $15^{15}$ is $3^{15} \times 5^{15}$. Counting the unique divisors by taking the combination of exponents for each prime factor and multiplying them together, we find that there are 22950 positive integers that are divisors.
Step-by-step explanation:
To find the number of positive integers that are divisors of at least one of $12^{12}$, $10^{10}$, and $15^{15}$, we need to find the prime factorizations of these numbers and count the unique divisors.
Prime Factorization:
The prime factorization of $12^{12}$ is $2^{24} \times 3^{12}$, the prime factorization of $10^{10}$ is $2^{10} \times 5^{10}$, and the prime factorization of $15^{15}$ is $3^{15} \times 5^{15}$.
Counting Unique Divisors:
We count the unique divisors by taking the combination of exponents for each prime factor and multiplying them together.
For $2$, we have $24+10 = 34$ possible combinations of exponents
For $3$, we have $12+15 = 27$ possible combinations of exponents.
For $5$, we have $10+15 = 25$ possible combinations of exponents.
Multiplying these counts together, we get $34 \times 27 \times 25 = 22950$.
Therefore, there are 22950 positive integers that are divisors of at least one of $12^{12}$, $10^{10}$, and $15^{15}$.