Divisibility & Primes, Ch 1, Q5

[apwiedmann]

I have a question about the very first chapter of Divisibility & Primes.

Question 5 states that j is divisible by 12 and 10 and asks the question whether it is divisible by 24.

What I do is that I prime factorise 12 and 10 and put all the prime factors in a partial prime box. This is what I have been taught to do in the chapter. Since I can build factors out of the prime factors, I multiply 2*2*2*3=24 and state that 24 is a factor.

Of course I see that with a numerical example 24 does not necessarily have to be a factor. But I do not get why one of the 2's may be redundant. From what information in the text am I supposed to know that one 2 may be redundant?

[cpallaka]

you should look at this information: divisible by 12 and 10.
one of ways to know what factors are redundant is
J is divisible by 12, so prime factor are 2, 2, 3
again J is also divisible 10, so prime factors are 2, 5.
in above, first statement has two 2's and 2nd statement has one 2. so you can say one 2 can be redundant.

another example: J is divisible by 12 and 18
J is divisible by 12, so prime factor are 2, 2, 3
again J is also divisible 18, so prime factors are 2,3,3.
in above, first statement has two 2's and 2nd statement has one 2. so you can draw one 2 can be redundant. again first statement has one 3 and 2nd statement has two 3's. so you can say one 2 and one 3 are redundant.

I hope this helps

[tim]

one thing to keep in mind about prime boxes is that you should never include more in a box than you absolutely need. you find out a number is divisible by 12, so you can put 2 2 3 in the prime box. then you find out it is also divisible by 10, so you ask yourself what MORE you need in the box. you already have a 2, so just put in a 5 and now your prime box has 2 2 3 5 (and possibly more, but these are the numbers you know). this set of numbers can build a 12 and also a 10, but we don't know if we can build a 24..

[afvatcha]

Somebody else posted this when I asked this question (I hope it's ok I am reposting someone elses answer):

Written by:tylrhllnbch


Picture a small, one-room building. This building has two windows in it, on different sides of the building. You can't see in either window. Susie goes up to Window 1, looks in, and comes over to tell you that there are a 2, a 2, and a 3 inside that room. (2, 2, and 3 are the prime factors of 12).

So what do you know? There are a 2, a 2, and a 3 in that room.

Then Amy goes up to Window 2, looks in, and comes over to tell you that there are a 2 and a 5 inside that room (2 and 5 are the prime factors of 10).

So what do you know? There are a 2 and a 5 in that room.

Now think about what you definitely know based on the info from BOTH Susie and Amy. There's definitely a 3 in the room. There's also a 5. What about the 2's? Are there definitely three 2's in there? Or could Susie and Amy have been looking at the same 2? We know there are at least two 2's in the room, because Susie reported two separate 2's. But that one 2 that Amy saw - that could have been one of the 2's that Susie saw. So I don't know for sure that there is a 3rd 2 in the room - there might be, but I just don't know.

What's that amount to? 2, 2, 3, 5 - just like we figured out from trying numbers. In other words, each statement gives us true BUT potentially overlapping information - and we have to strip out the overlap when we combine the statements.

[jnelson0612]

That's a great analogy afvatcha, and exactly right. Thanks for posting!

[afvatcha]

That's a great analogy afvatcha, and exactly right. Thanks for posting!

Just to be completely safe...and honest....I DON NOT come up with that one, but thanks!

[tim]

:)