jhjl121 wrote:

The function & represents the difference between the greatest and the least odd factors of a positive integer. If P and Q are positive integers, is &P > &Q?

(1) P is divisible by 45 but not by 75 and Q is divisible by 75 but not by 45

(2) Both P and Q have 6 odd factors

Now 45 = 3^2 * 5 and 75 = 3 * 5^2. 45 has two 3's and one 5. 75 has one 3 and two 5's.

Statement 1:

P is divisible by 45 but NOT by 75. This means there are at least two 3's in P but only one 5 (because if there is more than one 5 then it will become divisible by 75 also).

So P is at least = 3^2 * 5. Now there could be other prime numbers in P like 2, 7, 11 etc with various powers we dont know. Please note that the least odd factor of P is '1'.

Q is divisible by 75 but NOT by 45. This means there are at least two 5's in P but only one 3 (because if there is more than one 3 then it will become divisible by 45 also).

So P is at least = 3 * 5^2. Now there could be other prime numbers in P like 2, 7, 11 etc with various powers we dont know. Here also the least odd factor is '1'.

Since we dont know about other prime numbers in P & Q, we cant say what would be greatest odd factor of either P or Q. Eg. if P = 3^2 * 5 * 7 then the greatest odd factor of P will be 3^2 * 5 * 7 (P itself). So we cannot say anything about the asked question here.

Not Sufficient.

Statement 2:

Both have 6 odd factors. A number with 6 odd factors has its prime factorisation of the form: either 2^n * p^5 or 2^n * p1^2 * p2 (where p or p1/p2 are odd prime numbers and n could be any non negative integer. Increasing powers of 2 wont make any increase in number of odd factors because 2 is an even number). We dont know which of these forms P/Q take, we also dont know about which odd prime numbers are present in P/Q so we cannot say anything about the asked question here.

Not Sufficient.

Combining the statements:

We know P is at least 3^2 * 5. 3^2 * 5 has exactly 6 odd factors (1, 3, 3^2, 5, 3*5, 3^2*5).

Now no matter how many 2's we add to P, it wont change the number of odd factors. But if we add another odd prime number or even increase the power of 3 in P, it will increase the number of odd factors of P. Since there are only 6 odd factors, this means P is of the form = 3^2 * 5 * 2^n (where n is any non negative integer)

Here we should note that P has exactly 6 odd factors and the highest odd factor of P is 3^2 * 5 or '45'. So &P = 45-1 = 44.

Now Q. We know Q is at least 3 * 5^2. 3 * 5^2 has exactly 6 odd factors (1, 3, 5, 5^2, 3*5, 3*5^2).

Now no matter how many 2's we add to Q, it wont change the number of odd factors. But if we add another odd prime number or even increase the power of 5 in Q, it will increase the number of odd factors of Q. Since there are only 6 odd factors, this means Q is of the form = 3 * 5^2 * 2^n (where n is any non negative integer)

Here we should note that Q has exactly 6 odd factors and the highest odd factor of Q is 3 * 5^2 or '75'. So &Q = 75-1 = 74.

Now the question can be answered.

Sufficient. Thus

C is our answer.