In how many ways can the number 18900 can be split in two factors which are relative prime (or co-prime).

To find the number of ways the number 18900 can be split into two factors that are relatively prime (or co-prime), we need to follow these steps:

Step 1: Prime Factorization of 18900
We begin by finding the prime factorization of 18900.

First, divide by 2 (the smallest prime number): 18900 ÷ 2 = 9450
9450 ÷ 2 = 4725 (since 9450 is even, divide by 2 again)
4725 ÷ 3 = 1575 (since 4725 is divisible by 3)
1575 ÷ 3 = 525 (since 1575 is divisible by 3)
525 ÷ 3 = 175 (since 525 is divisible by 3)
175 ÷ 5 = 35 (since 175 is divisible by 5)
35 ÷ 5 = 7 (since 35 is divisible by 5)
7 ÷ 7 = 1 (since 7 is a prime number)

So, the prime factorization of 18900 is: 18900 = 2² × 3³ × 5² × 7

Step 2: Understanding Relative Primeness
Two numbers are relatively prime (co-prime) if they have no common prime factors. This means that when we split 18900 into two factors, one factor must contain some primes and the other factor must contain the rest, with no overlap in their prime factors.

Step 3: Distributing the Prime Factors
We have the prime factorization: 18900 = 2² × 3³ × 5² × 7.
We need to split these prime factors into two groups, ensuring that no prime factor is shared between the two groups.

For the factor 2², we can place both 2's in one group or in the other group. So, there are 2 choices (either all in one factor or all in the other factor).
For the factor 3³, we can place all three 3's in one group or in the other group. So, there are 2 choices (either all in one factor or all in the other factor).
For the factor 5², we can place both 5's in one group or in the other group. So, there are 2 choices (either all in one factor or all in the other factor).
For the factor 7, we can place the 7 in one group or in the other group. So, there are 2 choices (either in one factor or in the other factor).
Step 4: Calculating the Total Number of Ways
Since the choices for each prime factor are independent, we multiply the number of choices for each factor:

Total number of ways = 2 (choices for 2²) × 2 (choices for 3³) × 2 (choices for 5²) × 2 (choices for 7)

So, the total number of ways = 2⁴ = 16.

Step 5: Correcting for Repetition
The above 16 ways include both (a, b) and (b, a) as distinct pairs, but these should be considered the same since the order doesn't matter. Thus, we divide by 2 to avoid counting the same factor pair twice.

Therefore, the total number of distinct ways to split 18900 into two factors that are relatively prime is: 16 ÷ 2 = 8.

Final Answer:
There are 8 ways to split the number 18900 into two factors that are relatively prime.