5lbs. Chapter 13: Divisibility and Primes Problem 38

[xerocoool]

Q) If n is an integer and n^3 is divisible by 24, what is the largest integer number that must be a factor of n ?

A) 1
B) 2
C) 6
D) 8
E) 12

For n^3 to be divisible by 24 => it should have all the prime factors of 24 (i.e. 2x2x2x3)

So eliminating choices A,B,D

6 => 2X3 :: n^3 => 2^3 * 3^3 (which will be divisible by 24)
12 => 2x2x3 :: n^3 => 2^6 * 3^3 (which will also be divisible by 24)

I am unable to understand the latter part of the question. The largest number that has to be a factor of n has to be 12 right ? (the answer says 6). 6 would have been right if the question stated "what is the number that must be a factor of n".

Please correct me if I am wrong :)

Thanks
-Xeroo

[tommywallach]

Hey Xero,

I think you're thinking of the math right, but getting the final question wrong.

The largest thing that MUST be in n is 6. For example:

If I do 6^3, I get 216.

216 IS divisible by 24. (It's 9*24).

So clearly, n did not have to be divisible by 12. The largest thing that MUST go into n is 6.

-t

[xerocoool]

between n = 6 and n = 12

"The largest integer that is a factor of n" => does it imply the GCF of 12 and 6 => 6. (6 being the highest factor common to both)

Thanks
-xero

[tommywallach]

Hey Xerox,

Exactly right.

-t

[xerocoool]

Thanks :)

[tommywallach]

Glad to help!

-t