120 can be written as 2^p x q x r where p,q and r are prime numbers find the value of each p,q and r

Answer 1

if 120 can be represented as 2^p x q x r, how do we find p,q,r...

We also know that p,q,r are all prime...

Start with 120/(q x r) --> The maximum of q x r could be 120 but that would leave p = 0 which is not considered prime.

So at worst p = 1 --> 2 = 120/(q x r) --> q x r = 60

Can two prime numbers be found that multiply to 60? We now find the prime representation for 60...

60 = 2 * 30 = 2 * 2 * 15 [ cannot be two prime numbers ]

60 = 3 * 20 = 3 * 4 * 5 [ cannot be two prime numbers ]

60 = 5 * 12 = 5 * 2 * 6 [ cannot be two prime numbers ]

So 60 is not the number and p ≠ 1.

If p = 2 --> 4 = 120/ (q x r) --> q x r = 30

30 = 2 * 15 = 2 * 3 * 5 [ cannot be two prime numbers ]

30 = 3 * 10 = 3 * 2 * 5 [ cannot be two prime numbers ]

30 = 5 * 6 = 5 * 3 * 2. [ cannot be two prime numbers ]

So 30 is not the number and p ≠ 2.

If p = 3 --> 8 = 120/ (q x r) --> q x r = 15

15 -= 3 * 5 --> solution where q and r are both prime...

Therefore p = 3, (q,r) = (3,5) or (q,r) = (5,3)