The greatest common factor of 16

[ajafari]

The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
3
14
30
42
70

OA 42

The explanation states that a 7 but not a 5 could be a common factor. Can someone please explain this. I don't see how 7 is a common factor.

[vili_exisu]

Because the question states that the GCD between n and 45 is only 3. Thus, 5 cannot be a factor of n, but 7 could be.

[esledge]

This is almost a verbal question! ajafari, your question boiled down to the difference between is and could.

I don't see how 7 is a common factor.

Because the question states that the GCD between n and 45 is only 3. Thus, 5 cannot be a factor of n, but 7 could be.

[bpriya]

Shouldn't 42, the answer, and 16 still have a common factor of 4? Since 45 and 42 have a common factor of 3. Am I missing something?

[loving.achin]

@bpriya :-
>> Shouldn't 42, the answer, and 16 still have a common factor of 4? Since 45 and 42 have a common factor of 3. Am I missing something?

Nice question, but a small conceptual problem here. They are asking you GCF of n and 210. You can take this question in the following way.

GCF (16, n) = 4. This means that n is a multiple of 4 but not of 8 or 16. (this is quite important point)
GCF (45, n) = 3. This means that n is a multiple of 3 but not of 5 or 9 or 15 or 45. It comes from the following logic.
45 = 5 * 3 * 3
n and 45 has GCF = 3. This means that only 3 is the only common number between both and hence no other multiple of it exist in other number. Right?

Hence we got that 4 and 3 exists between a number n.

=> n = a multiple of 4 and 3.

Now GCF(n, 210) = ?
210 = 7 * 3 * 5 * 2
n = x * 4 * 3

The common number between the two are :
2 * 3.
But 6 is not in answer choice. This means there is something else also common between the two.

Can x be a multiple of 5 ? NO, as we ruled out 5 in case (2). i.e. GCF(n, 45)
Can x be a multiple of 2 ? NO, 210 has only one 2.
Can x be a multiple of 7 ? YES, as we haven;t ruled out any case in which 7 is not valid.

Hence x is a multiple of 7. Hence GCF = 2 * 3 * 7 = 42.


I hope it clears your doubt.


@ ajafari :- Did you see how we derived it to be a multiple of 7 above.

Please let me know if you have any other query/issue.

Thanks
Achin

[mschwrtz]

That looks good loving.achin. Another way to say the same thing:

Among the prime factor of n will be exactly two 2s (since GCF of n and 16 is 4, not 8, etc.), exactly one 3 (since GCF of n and 45 is 3, not 9, etc.) and exactly zero 5s (since GCF on n and 45 is not a multiple of 5). Anything else is permitted. Nothing else is required.

[suskom]

Not sure if I am missing something here but why couldn't the value of n be 12?

If that is the case, the greatest common factor of n (12) and 210 COULD be 3.

[mondegreen]

Not sure if I am missing something here but why couldn't the value of n be 12?

If that is the case, the greatest common factor of n (12) and 210 COULD be 3.

\When x and y are 2 integers, their GCD h is defined when

x/h = Integer and y/h = Integer.

The GCD of 12 and 210, as you say can be 3.

Indeed, 12/3 = 4 and 210/3 = 70. However, GCD stands for "GREATEST" common factor. What if I could find another integer, greater than 3, which still divides into 12 and 210 evenly?

The GCD of 12 and 210 is actually 6.

[RonPurewal]

Not sure if I am missing something here but why couldn't the value of n be 12?

If that is the case, the greatest common factor of n (12) and 210 COULD be 3.

"Greatest" means ... well, greatest.

3 is a common factor of 12 and 210, but it's not the greatest one. (See the post above this one.)