Question

How do you find the GCF of $ 15c{{d}^{2}},25{{c}^{2}}d $ ?

Answer 1

Hint: GCF means greatest common factor. GCF of 2 numbers is the greatest number that divides both the numbers. We can find the GCF of 2 numbers by writing the number as a product of its prime factor. We will apply this method to solve the above question.

Complete step by step answer:
We have to find the GCF of $ 15c{{d}^{2}} $ and $ 25{{c}^{2}}d $
We will write these numbers as a product of their prime factors and take all the common factors out. Product of all common factors will give the GCF of 2 numbers. In this case, we don’t know if c and d are prime numbers. To find out GCF c and d should be coprime that means there should not be any common factor between c and d except 1 otherwise the question is invalid due to less information we need more information about c and d.

Let’s write both numbers as a product of its prime. We don't know the prime factor of c and d so we will write as it is and assume b and c are co-prime.
 $ 15c{{d}^{2}}=3\times 5\times c\times d\times d $
 $ 25c{{d}^{2}}=5\times 5\times c\times c\times d $
We can see that common factors are 5,c,and d so the GCF of $ 15c{{d}^{2}} $ and $ 25{{c}^{2}}d $ is $ 5cd $ where b and c are co-prime. If they are not coprime we can see there is one d left in $ 15c{{d}^{2}} $ and one c left in $ 25{{c}^{2}}d $ there may be common factors between c and d so we need more information about c and d.

Note:
 Another method to find out GCF is to find the LCM of 2 numbers then divide it with a product of 2 numbers. LCM is the lowest common multiplier of 2 numbers. Keep in mind that in the above question we don’t know whether c and d have any common factors so we assumed they are co-prime.