The function PF is defined as PF(a) = k, where k is the number of prime factors of positive integer a. If PF(x) = PF(2x)=PF(3x) = 2 and PF(y) = PF(5y) = PF(7y) = 2, where x and y are positive integers, what is the value of PF(least common multiple of x and y)?a)2b)3c)4d)5e)Cant be determinedCorrect answer is option 'C'. Can you explain this answer?

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The function PF is defined as PF(a) = k, where k is the number of prime factors of positive integer a. If PF(x) = PF(2x)=PF(3x) = 2 and PF(y) = PF(5y) = PF(7y) = 2, where x and y are positive integers, what is the value of PF(least common multiple of x and y)?

a)
2
b)
3
c)
4
d)
5
e)
Can't be determined

Correct answer is option 'C'. Can you explain this answer?

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The function PF is defined as PF(a) = k, where k is the number of prim...

Given

PF(a) = k, where k is the number of prime factors of positive integer a
PF(x) = PF(2x)=PF(3x) = 2, where x is an integer > 0
PF(y) = PF(5y) = PF(7y) = 2, where y is an integer > 0

To Find: PF(LCM(x, y)?

Approach

PF(LCM(x, y) = Number of prime factors of LCM(x, y)
1. LCM(x, y) will have the same prime factors as x and y
2. As PF(x) = 2 and PF(y) = 2, LCM(x, y) can have:
  1. a maximum of 4 prime factors if the prime factors of x and y are distinct
  2. a minimum of 2 prime factors if the prime factors of x and y are same
For finding the prime factors of LCM(x, y), we need to know the prime factors of x and y
Finding prime factors of x
1. As PF(x) = PF(2x) = 2, i.e. both x and 2x have 2 prime factors
2. As PF(x) = PF(3x) = 2, i.e. both x and 3x also have 2 prime factors
3. We will use the above 2 relations to find the prime factors of x
Finding prime factors of y
1. As PF(y) = PF(5y) = 2, i.e. both y and 5y have 2 prime factors
2. As PF(y) = PF(7y) = 2, i.e. both y and 7y have 2 prime factors
3. We will use the above 2 relations to find the prime factors of y
Once we find the prime factors of x and y, we can calculate the prime factors of LCM(x, y)

Working Out

Finding prime factors of x
1. As PF(x) = PF(2x) = 2, i.e. both x and 2x have 2 prime factors and 2 is one of the prime factors of x
2. As As PF(x) = PF(3x) = 2, i.e. both x and 3x also have 2 prime factors and 3 is one of the prime factors of x
3. Thus, x has 2 and 3 as its prime factors.
Finding prime factors of y
1. As PF(y) = PF(5y) = 2, i.e. both y and 5y have 2 prime factors and 5 is one of the prime factors of y.
2. As PF(y) = PF(7y) = 2, i.e. both y and 7y have 2 prime factors and 7 is one of the prime factors of y.
3. Thus, y has 5 and 7 as its prime factors.
As x and y do not have any common prime factor, LCM(x, y) will have {2, 3, 5, 7} as its prime factors.
Thus PF(LCM(x, y)) = 4

Answer: C

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