Castor.kim wrote:This is simple but not easy factor problem..
certainly, these problems no so hard to solve, but it takes too much time to solve..

how can I decrease the time spending.. in these problem?

* is "n" a factor of 24 ?

1) 3n is a factor of 48
2) 5n is a factor of 60

OA is B

1) Since any factor of 24 will also be a factor of 48, we really only have to look at the factors that are unique to 48: 16 and 48. 3n cannot be 16 because n would then be a fraction. 3n could be 48, would would make n 16. 16 is not a factor of 24. We can easily see that n could be a factor of 24; if 3n=6, then n=2. Insufficient.

2) Here, we only have to look at factors of 60 that are multiples of 5: 5, 10, 15, 20, 30, and 60. If we factor out a 5 to leave us with n, we can see that n could be 1, 2, 3, 4, 6, or 12, all of which are factors of 24. Sufficient.

Hi Kim,

The timing used to be a big challenge for me on these problems as well. The reason was that my go-to strategy was picking numbers. While that strategy is often quite effective, a problem with larger numbers like this one can make it too time consuming. What do we do?

It turns out the world of divisibility is quite similar to the world of Transformers.... primes are all powerful! They will be our key to moving through divisibility problems with a consistent and quick process.

'Is n a factor of 24?'

Let's get to our primes! 24 = 2*2*2*3. For n to be a factor of 24, n can have no more than three 2s and one 3; it can have nothing else. The question translates to "Does n have three or fewer 2s and one or fewer 3s in its prime factors?'

1) 3n is a factor of 48.... it's prime time! 48 = 2*2*2*2*3
So 3n can have no more than four 2s and no more than one 3.
Therefore, this statement translates to "n has four or fewer 2s."
Not Sufficient

2) 5n is a factor of 60... primes! 60 = 2*3*5
So 5n can have no more than one 2, one 3, and one 5.
Therefore, this statement translates to "n has no more than one 2 and one 3."
Sufficient

Answer B

We don't often think of numbers as a product of their primes, so it can feel unnatural at first. However, it's a very useful approach to a wide array of questions, and with a little practice, it's quite fast as well.