Deduction on data sufficiency questions

Have a look at a data sufficiency question in which we can use factors to deduce much more information than what we might first guess: When the digits of a two-digit, positive integer M are reversed, the result is the two-digit, positive integer N. If M > N, what is the value of M? Statement 1: The integer (M – N) has 12 unique factors. Statement 2: The integer (M – N) is a multiple of 9. With this question, we are told that M is a two-digit integer and N is obtained by reversing it. So if M = 21, then N = 12; if M = 83, then N = 38 (keeping in mind that M must be greater than N). In the generic form: M = 10a + b and N =10b + a (where a and b are single-digit numbers from 1 to 9. Neither can be 0 or greater than 9 since both M and N are two-digit numbers.) We also know that no matter what M and N are, M > N. Therefore: 10a + b > 10b + a 9a > 9b a > b Let’s examine both of our given statements: Statement 1: The integer (M – N) has 12 unique factors. First, let’s figure out what M – N is: M – N = (10a + b) – (10b + a) = 9a – 9b Say M – N = A. This would mean A = 9(a-b) = 3^2 * (a-b) The total number of factors of A where A = X^p * Y^q *… can be calculated using the formula (p+1)*(q+1)* … We know that A has 3^2 as a factor, so X = 3 and p = 2. Therefore, the total number of factors would be (2+1)*(q+1)*… = 3*(q+1)*… = 12, so (q+1)*… must be 4. Case 1: This means q may be 3 so that (q+1) is 4. Since a-b must be less than or equal to 9 and must also be the cube of a number, (a-b) must be 8. (Note that a-b cannot be 1 because then the total number of factors of A would only be 3.) So, a must be 9 and b must be 1 in this case (since a > b). The integers will be 91 and 19, and since M > N, M = 91. Case 2: Another possibility is that (a-b) is a product of two prime factors (other than 3), both with the power of 1. In that case, the total number of factors = (2+1)*(1+1)*(1+1) = 12 Note, however, that the two prime factors (other than 3) with the smallest product is 2*5 = 10, but the difference of two single-digit positive integers cannot be 10. This means that only Case 1 can be true, therefore, Statement 1 alone is sufficient. This is certainly not what we expected to find from just the total number of factors! Statement 2: The integer (M – N) is a multiple of 9. M – N = (10a + b) – (10b + a) = 9a – 9b, so M – N = 9 (a-b) . This is already a multiple of 9. We get no new information with this statement; (a-b) can be any integer, such as 2 (a = 5, b = 3 or a = 7, b = 5), etc. This statement alone is insufficient, therefore our answer is A.