Number Properties-Divisibility & Primes- Problem 4- Page 23

[jasonsai9]

Question states:
Given that 8 is not a factor of g, is 8 a factor of 2g?

The answer key states, In order for 8 to be a factor of 2g, we would need two mor 2's in the prime box, and by the factor foundation rule , g would need to be divisible by 4, we know that g is not divisible by 8, but there are certainly integers such as 4,12,28 etc that are divisible by 4 and not by 8.

"However while we cannot conclude that g is Not divisible by 4, we cannot be certain that g is divisible by 4 either"

-- I am confused, my question is dont these two sentences mean the same that basically g is divisible by 4?? Please advise.

[georgepa]

Question states:

"However while we cannot conclude that g is Not divisible by 4, we cannot be certain that g is divisible by 4 either"

-- I am confused, my question is dont these two sentences mean the same that basically g is divisible by 4?? Please advise.

What it means is the we cannot say anything about the divisibility of g by 4. Meaning you can pick g in such a way 8 is a factor of 2g in some cases and in other cases it is not.

The example they give:

g = 5 => 2g = 10 => 8 is not a factor of 2g
g = 4 => 2g = 8 => 8 is a factor of 2g

[osaisai]

Thank You Georgepa!!!!

[esledge]

Thanks George!

[francisat]

Hello - I am puzzled as to why we focus on the divisibility of g by 4 in the second sentence of the explanation: "By the Factor Foundation Rule, g would need to be
divisible by 4 "

My question is why do we talk about the requirement that g would need to be divisible by 4 and why not instead talk about 2g divisibility by 4 ? I'm confused about that part of the answer and don't understand the link and transition from 8 / 2g to 4 / g. Thank you for help.

[Ben Ku]

Hello - I am puzzled as to why we focus on the divisibility of g by 4 in the second sentence of the explanation: "By the Factor Foundation Rule, g would need to be
divisible by 4 "

My question is why do we talk about the requirement that g would need to be divisible by 4 and why not instead talk about 2g divisibility by 4 ? I'm confused about that part of the answer and don't understand the link and transition from 8 / 2g to 4 / g. Thank you for help.

In order for 2g to be divisible by 8, 2g needs to be divisible by 2 three times. Since it already has a 2, then g needs to be divisible by 2 two times; g needs to be divisible by 4.

Using prime boxes, we want to know if 2g has three 2's in its prime box. It already has one 2, so we just need to see if g has two 2's.

Hope that makes sense.

[francisat]

Hello Ben many thanks for this explanation - yes it now makes perfect sense.

Many thanks for your help.

Regards

Frank

[Ben Ku]

Great! Glad it helped.

[rockrock]

can you pls tell me if i am thinking about this correctiy for a different, but similar problem?

from the strategy guide, number properties page 23 #10

If k is divisible by 5 and 3k is not divisble by 5, then is K divisible by 10?

i would think the easiest way to think of this that since the prime factors of 10 are 2 and 5...the only way for 3k to be divisble by 10, is if it includes these two prime factors, which the question already tells us it doesn't include 5.

i thought the book explanation was a bit confusing...

[tim]

First, problem 10 says k is divisible by 6, not by 5. If you use 5 the problem is invalid..

If 3k is not divisible by 5, then k is certainly not divisible by 5 - think of it this way: if a 5 doesn't show up in the prime box for 3k, it can't show up in either the 3 compartment or the k compartment (remember if you have a prime box for a product like 3k you can split it into two separate compartments). If 5 isn't in k's prime box, there's no way you can build a 10, because a 10 requires both a 2 and a 5..

Be careful reading problems like this BTW; not only did you transcribe a 6 as a 5, but you also seem to have gotten 3k and k confused at some point in your answer..