total number of divisors of 75600 which are divisible by 3 is

To find the total number of divisors of 75600 that are divisible by 3, we first need to determine the prime factorization of 75600. This will allow us to understand its divisors better.

Prime Factorization of 75600

We can start by breaking down 75600 into its prime factors. Let’s do this step-by-step:

• 75600 is even, so we can divide by 2:
• 75600 ÷ 2 = 37800
• 37800 is also even, so we divide by 2 again:
• 37800 ÷ 2 = 18900
• 18900 ÷ 2 = 9450
• 9450 ÷ 2 = 4725 (now we switch to the next prime since 4725 is odd)
• 4725 is divisible by 3: 4725 ÷ 3 = 1575
• 1575 is also divisible by 3: 1575 ÷ 3 = 525
• 525 ÷ 3 = 175 (now we switch to the next prime)
• 175 is divisible by 5: 175 ÷ 5 = 35
• 35 is divisible by 5: 35 ÷ 5 = 7
• 7 is a prime number.

Putting all of this together, we have:

75600 = 2^4 × 3^3 × 5^2 × 7^1

Calculating the Total Divisors

The formula for calculating the total number of divisors from the prime factorization is:

(e1 + 1)(e2 + 1)(e3 + 1)... where e1, e2, e3, etc., are the powers of the prime factors.

For our factorization:

• For 2: e1 = 4 → (4 + 1) = 5
• For 3: e2 = 3 → (3 + 1) = 4
• For 5: e3 = 2 → (2 + 1) = 3
• For 7: e4 = 1 → (1 + 1) = 2

Now, multiplying these results together gives:

Total Divisors = 5 × 4 × 3 × 2 = 120

Divisors Divisible by 3

Next, to find the divisors that are divisible by 3, we need to consider the prime factorization excluding the factor of 3. This means we will look at:

2^4 × 3^2 × 5^2 × 7^1

Now we adjust the power of 3 down by one because we want to count the divisors that include at least one factor of 3:

• For 2: 4 → (4 + 1) = 5
• For 3: 2 → (2 + 1) = 3
• For 5: 2 → (2 + 1) = 3
• For 7: 1 → (1 + 1) = 2

Multiplying these gives:

Divisors Divisible by 3 = 5 × 3 × 3 × 2 = 90

Final Result

Thus, the total number of divisors of 75600 that are divisible by 3 is 90. This shows how the prime factorization and divisor counting principles work together to provide a complete picture of the number's divisors.