Problem Solving

For all integers p, <<p>> is the product of p’s distinct prime factors. What is the greatest common factor of <<24>> and <<54>>?

a. <<2>>
b. <<3>>
c. <<8>>
d. <<12>>
e <<14>>

How do I figure this out?

richs_ca wrote:For all integers p, <<p>> is the product of p’s distinct prime factors. What is the greatest common factor of <<24>> and <<54>>?

a. <<2>>
b. <<3>>
c. <<8>>
d. <<12>>
e <<14>>

How do I figure this out?

Break it down!

54 = 2 * 27 = 2*3*3*3
So, the distinct prime factors of 54 are 2 and 3, and <<54>> is 2*3 = 6

24 = 2*12 = 2*2*2*3
So, <<24>> is also 2*3 = 6

Therefore, the GCF of <<24>> and <<54>> is 6.

Now, the answers are also in <<x>> form, so we want to pick a choice that's made up of 2s and 3s.

a) no 3
b) no 2
c) no 3
d) only 2s and 3s! Winner!
e) no 3

choose (d).

The answer is d. <<12>>

So you know the definition of <<p>>.
To find <<24>>, break 24 in prime factors. 24 = 2*2*2*3. 2 and 3 are the only distinct prime factors and there product is 6. So <<24>> = 6
Similarly for <<54>>, 54 = 2*3*3*3 and <<54>> = 6

LCM of 6 & 6 is 6. So if the answers were provided in simple integers the answer would be 6. But the answer is a multiple of 6 or should only contain 2 and 3 as the only prime factors.

Looking at the answers <<12>> is the only answer that matches the criteria.