Division - One puzzle a day - Puzzle Buddies

Puzzle:Find the number of factors of 264,600 that are not divisible by 6.
Note: Don't forget to visit us again. Answer to the puzzle will be posted tomorrow.
Solution: 264,600 = 2^3*3^3*5^2*7^2 (broken into prime numbers)

There are 4 powers of 2: 2^0,2^1,2^2 and 2^3

There are 4 powers of 3: 3^0,3^1,3^2 and 3^3

There are 3 powers of 5: 5^0,5^1, and 5^2

There are 3 powers of 7: 7^0,7^1, and 7^2

There are 4*4*3*3=144 different ways to combine the prime factors of 264,600.

Now, eliminate the factors of 6. A factor of 6 must have at least one 2 and one 3.

So it must either have either 2^1,2^2, or 2^3 AND 3^1,3^2, or 3^3. So, there are 3*3=9 different ways the powers of 2 and 3 can combine to generate distinct numbers divisible by 6.

Also, there are 3*3 = 9 different numbers that can be created from the powers of 5 and 7. Any of these 9 numbers can combine with any of of the 9 multiples of 6 to form 9*9 = 81 distinct multiples of 6.

Hence, the number of factors of 264,600 that are no divisible by 6 is 144-81 = 63.
Winner: Noone