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03 Jun 2020, 09:34
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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
Are You Up For the Challenge: 700 Level Questions
A. 2
B. 3
C. 4
D. 5
E. Can't be determined
Are You Up For the Challenge: 700 Level Questions
Bunuel
Given:The function PF is defined as PF(a) = k, where k is the number of prime factors of positive integer a.
Asked: 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)?
PF(x) = PF(2x) = PF(3x) = 2; \(x = 2^x*3^y \)
PF(y) = PF(5y) = PF(7y) = 2; \(y = 5^w*7^z \)
\(LCM(x,y) = 2^x*3^y*5^w*7^z\)
PF(LCM(x,y)) = 4
IMO C
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03 Jun 2020, 10:08
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Bunuel
PF(a) = k, where k is the number of prime factors of positive integer a.
If PF(x) = PF(2x) = PF(3x) = 2
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)?
PF(x) = PF(2x) = PF(3x) = 2, implies 2 prime factors of number x, well,
concept: note that in above equation we have x, 2x, and 3x and all 3 have 2 prime factors, also note 2x and 3x have factors of 2 and 3 as prime numbers
one thing to understand is if we take any number having prime number other than 2, 3 as factors then, we are going to add up number of prime factors for 2x and 3x and thus the equation will not hold.
lets start with prime numbers, each prime number has only 1 prime factor, that is itself.
and 1 is not a prime number
so, x cant be a prime number, well x in not a prime number.
x can be then, 4, 6, 8, 9, 10, 12.........
but given: PF(x) = PF(2x) = PF(3x) = 2
or x has two different prime numbers as factor, lets assume x has a prime numbers as factor as 2, 3
x= 6, 12, 18,..........
x= 2*3, 2^2*3, 2*3^2,.....
x has 2 factors as prime numbers
lets try for PF(2x) =2 or 2x has also 2 factors as prime numbers
2x = 12, 24, 36,....
2x = (2^2)*3, (2^3)*3, (2^2)*(3^2),.......
so, 2x has indeed 2 prime numbers, the same as x, as factors.
lets try for PF(3x) = 2, or 3x has also 2 prime numbers as factors.
3x= 18, 36, 54,....
3x = 2*(3^2), (2^2)*(3^2), 2*3^3,.....
so, 3x has 2 factors as prime numbers.
therefore, x=6, 12, 18,......
PF(y) = PF(5y) = PF(7y) = 2
again, we see 5y and 7y have prime factors, as 5 and 7; therefore, the best place to start y is to use these two prime factors.
therefore,
y= 35, 35*5, 35*7, 35*35,......
each of the y has exactly 2 prime factors as 5, 7.
now whats asked is PF(LCM of x and y),
x = 6, 12, 18,,.......
y= 35, 35*5, 35*7, 35*35,......
how to find a LCM
The least common multiple, or LCM, is another number that's useful in solving many math problems. Let's find the LCM of 30 and 45. One way to find the least common multiple of two numbers is to first list the prime factors of each number.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs.
2: one occurrence
3: two occurrences
5: one occurrence
2 × 3 × 3 × 5 = 90 <— LCM
After you've calculated a least common multiple, always check to be sure your answer can be divided evenly by both numbers.
coming back to question:
x = 6, 12, 18,,.......
y= 35, 35*5, 35*7, 35*35,......
with so many possibilties of x and y there can be infinite LCM, but all LCM will have same 4 prime numbers as factors, that is 2, 3, 5, 7.
therefore, PF(LCM of x and y) = 4
