find the number of factors of 2^8 × 7^5 × 14
Find the number of factors of 2^8 × 7^5 × 14
To find the number of factors of a given number, we need to determine the prime factorization of the number and then apply a formula.
Let’s start by finding the prime factorization of the given number: 2^8 × 7^5 × 14.
The prime factorization of 2^8 can be written as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256.
The prime factorization of 7^5 can be written as 7 × 7 × 7 × 7 × 7 = 16,807.
The prime factorization of 14 can be written as 2 × 7.
Therefore, the prime factorization of 2^8 × 7^5 × 14 is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7 × 7 × 7 × 7 × 7 × 2 × 7.
Now, let’s find the number of factors using the formula:
If the prime factorization of a number is written as p1^a1 × p2^a2 × p3^a3 × … × pn^an, where p1, p2, p3, …, pn are distinct prime numbers, and a1, a2, a3, …, an are positive integers, then the number of factors is given by (a1 + 1) × (a2 + 1) × (a3 + 1) × … × (an + 1).
In our case, the prime factorization of 2^8 × 7^5 × 14 is 2^8 × 7^5 × 2 × 7.
Using the formula, the number of factors = (8 + 1) × (5 + 1) × (1 + 1) × (1 + 1) = 9 × 6 × 2 × 2 = 216.
Therefore, the number of factors of 2^8 × 7^5 × 14 is 216.