theGame001 wrote:
I am really sorry and I will understand if you don't reply.
Okay so I got that 264600 has 144 factors.
Just for my understanding lets say we have to find out how many numbers out of 144 are divisible by 6. After this step I am unable to understand why are we finding factors of 144?
No. We are not finding the factors of 144.
264600 has 144 factors. They are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, ... , 264600
Now some of them are divisible by 6 and some are not.
Divisible by 6: 6, 12, 18 ...
Not divisible by 6: 1, 2, 3, 4, 5, ...
Now how do you split them - how many are multiples of 6 and how many are not.
The common thing about the multiple of 6 is that they have 6 in them i.e. they have a 2 and a 3.
\(264600 = 2^3 * 3^3 * 5^2 * 7^2\)
If you have understood the method of calculating 144, you should easily be able to understand the way we calculate multiples of 6.
To get a multiple of 6, we need at least one 2 and at least one 3.
So you can select 2 in 3 ways (either one 2, two 2s or three 2s. You cannot have zero 2s since you need to make a 6)
You can select 3 in 3 ways (either one 3, two 3s or three 3s. You cannot have zero 3s since you need to make a 6)
You can select 5 in 3 ways (either zero 5, one 5 or two 5s)
You can select 7 in 3 ways (either zero 7, one 7 or two 7s)
Total number of multiples of 6 = 3*3*3*3 = 81
Total number of factors which are not multiples of 6 = 144 - 81 = 63
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