a)

If 6 is a factor of a number, that number is divisible by 6, and we get a whole number.

Dividing by 6 is the same as dividing by 3 first, then by 2. (Or by 2 first, then by 3)


So if dividing by 6 gives a whole number, then dividing by 3 only would give a result 2 times of that whole number (since you didn't further divide it by 2)

Likewise, dividing by 2 only would give a result 3 times of that whole number (since you didn't further divide it by 3)

Eg.


42 ÷ 6 = 7


But 42 ÷ 6

= 42 ÷ 3 ÷ 2

= 14 ÷ 2

= 7

And,


42 ÷ 6

= 42 ÷ 2 ÷ 3

= 21 ÷ 3

= 7

So since 42 is divisible by 6 , it is also divisible by 3 and by 2 since we get 14 and 21 respectively.


Both 3 and 2 and distinct prime factors that form the composite factor 6. They don't contain each other.



b)


Same logic but in the opposite way.


If 2 and 7 are factors, then the number is divisible by 2 to get a whole number and also divisible by 7 to get another whole number.

So,

½ of that number is a whole number
(let's call it a).

1/7 of that number is also a whole number (let's call it b)

Then 3/7 of that number is also whole (since b is whole then 3b must be whole as well)


½ of number - 3/7 of number = a - 3b

7/14 of number - 6/14 of number = a - 3b

1/14 of number = a - 3b

Since a is whole and 3b is whole, a - 3b must be whole as well.


So 1/14 of the number is whole. This means that the number is exactly divisible by 14 to give a whole number.



2 and 7 are distinct prime factors that don't contain each other.



c)


8 = 2 x 2 x 2 = 2³

The factor 8 already contains 2 inside. The factor 2 is a subset of the factors of 8.


So if a number is divisible by 8, we can at most say it is divisible by 4 as well (= 2 x 2) or is divisible by 2 as well.


But 16 = 2 x 2 x 2 x 2 = 2⁴


To be divisible by 16, the number must have at least 4 '2s' in its prime factorisation.


Eg. 96 = 2^5 x 3

= 2⁴ x 2 x 3

= 16 x 6



Some counter examples :


① 8 is divisible by both 8 and 2. But 8 is not divisible by 16 to get a whole number.


② 24 is divisible by both 8 and 2,
(24 ÷ 8 = 3, 24 ÷ 2 = 12)

But 24 is not divisible by 16 to get a whole number.


③ 40 is divisible by both 8 and 2

(40 ÷ 8 = 5, 40 ÷ 2 = 20)

But 40 is not divisible by 16 to get a whole number.


④ 72 ÷ 8 = 9, 72 ÷ 2 = 36

But 72 is not divisible by 16 to get a whole number.

a) 6 is factorised into 2 × 3, therefore if a number is divisable by 6 ( 6 is a factor ) it can be written as 6 × a which would be 2 × 3 × a therefore divisible by 3
Thus true

b) If 2 and 7 are a factor of a number, it can be written as 2 × 7 × a which = 14 × a, therefore divisible by 14
Thus true

c) If 2 and 8 are a factor of a number, why is 16 not a factor, 8 can be broken down into 2 ×2 ×2, thats where the factor 2 comes from. 8 is not a prime number therefore cannot be considered when doing prime factorisation. Best explained with counter example would be number 24.

I see, so question c is false because 8 is not a prime number and it comes from 2. Thank you so much for showing the counter examples and both of the thorough explanations to my question.

No, its not because 8 isn't prime.

Your two factors can be either composite or prime, as long as within them, they don't have overlapping prime factors with each other.

For c), it's because 8 already contains 2 in its prime factorisation. So we can only say it is divisible by 8 at most.



Here's an example :


If a number is divisible by 57 and divisible by 14, (both composite) it is divisible by (57 x 14) = 798

8778 is divisible by all three.


8778 ÷ 57 = 154
8778 ÷ 14 = 627
8778 ÷ 798 = 11


We note that 57 = 19 x 3 (both prime)
We note that 14 = 2 x 7 (both prime)


There are no overlapping prime factors within them.

One more


If 6 and 4 are both factors of a number, it doesn't necessarily mean it is divisible by 24 (= 4 x 6)


Eg.

36 ÷ 4 = 9, 36 ÷ 6 = 6
12 ÷ 4 = 3, 12 ÷ 6 = 2
60 ÷ 4 = 15, 60 ÷ 6 = 10


But 36 ,12 and 60 are not divisible by 24.


We note that 4 = 2² and 6 = 2 x 3


There is already one 2 that is overlapping, with the second 2 and the 3 not overlapping.


So at most we can say it is divisible by (2 x 3 x 4) = 12

I thank you again for correcting me, appreciate it a lot.