Hello, 8)

Hope all is going your way today. I am currently trying to resolve a wrinkle with the Divisibility exercises:

Chapter 1, #9 Yes While #2 is Can not be determined. What's the difference there?

Question #5: Easy question - discussion would be helpful
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Is 24 a factor of J? factors 10 are 2, 5 factors of 12 are 2, 2, 3

J factors per book
2, 2, 3, 5

why not 2, 2, 2, 3, 5?

Hi, I would be happy to help but need you to post the full text of any questions about which you ask - I don't always have access to all of my books when I'm posting (and, in any event, I wouldn't be able to get through everyone's questions if I also had to look up all of the problems). Thanks!

Full Question Is:

If j is divisible by 12 and 10, is j divisible by 24?

Book says CANNOT BE DETERMINED. If j is divisible by 12 and by 10, its prime factors include 2,2,3, and 5, as indicated by the prime box to the left. Therefore, any integer that can be constructed as a product of these factors is also a factor o j. 24 = 2 * 2 * 2 * 3. There are only two 2s in the prime box, therefore, 24 is not necessarily a factor of j.

I BELIEVE this might be a typo. The factors of 12 and 10 are 2*2*3*5*2, which can make 24. Can you confirm?

Howdy. You are definitely not the first person to make this assumption. The key to this problem is understanding the overlap of prime boxes. Let me begin with a simpler example.

If j is divisible by 2 and 4, is j divisible by 8?

In this case, the prime box of 2 = 2, and the prime box of 4 = 2*2. However, j does not necessarily have to be divisible by 8, as j could just as easily be 4 (a number divisible by both 2 and 4 but not divisible by 8).

The key is recognizing that prime boxes, when combined, and not additive. Instead, when prime boxes overlap in a problem such as this one, take the HIGHER POWER of the specific factor that overlaps. Thus in the problem at hand, the prime box of 12 = 2*2*3, and the prime box of 10 = 2*5. The combined prime box of j must include 3*5*2*2 (one 3, one 5, and the higher power of the shared factor 2). Since 24 = 3*2*2*2, we CANNOT determine whether j is divisible by 24.

To check your answer you could also consider actual numbers. If j is divisible by 12 and 10, j could = 60, which is not divisible by 24. J, however, could just as easily = 120, which is divisible by 24.

Hope that helps!
-dan

Can you explain question #4......Given that 8 is not a factor of g, is 8 a factor of 2g?

Thanks.

The explanation in our book uses prime boxes. That seems like a good way to think about the problem, but here's an alternative:

We are told that 8 is not a factor of g. So g could be any number that is not a multiple of 8.

For example, g could be 3. If g is 3, then 2g is 6. 8 is NOT a factor of 6.
Another possibility is that g could be 4. If g is 4, then 2g is 8. 8 IS a factor of 8.

From our examples, we can see that the given information (that 8 is not a factor of g) does not imply that 8 is, or is not, a factor of 2g.

-Jad

I am really confused as to when to use the prime box overlap principle and when not to, even I thought it was typo and that the answer was Yes.

In general, remember that when combining prime boxes, you can only include the MINIMUM possible. So if you're combining 12 and 10:
12 = 2*2*3
10 = 2*5

When I create a new box, the bare minimum I need to be able to construct both a 10 and a 12 is: 2*2*3*5. From those four numbers, I can create 12, and from those four numbers, I can also create 10.

Think about it this way:
You're standing outside a store. Inside the store is a big box of fruit. I go in, look in the box, come back out and tell you there are 2 apples and an orange in the box. Then someone else goes in, looks in the box, comes out, and tells you that there are an apple and a banana in the box.

Do you definitely know that there are 3 apples in the box? No - that second person have been talking about one of the apples that the first person was talking about. The most we can say definitively is that there are at least 2 apples in the box, not three. Same concept with the prime box - the 12 and the 10 are two separate pieces of info, and they could be using some of the same numbers from the prime box.