Divisibilty question - Number properties book

[mww7786]

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
'
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?

[StaceyKoprince]

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!

[Guest]

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?

[dbernst]

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

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?

[bryang]

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

Thanks.

[JadranLee]

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

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

Thanks.

[Guest]

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.

[StaceyKoprince]

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.