BTGmoderatorDC wrote: ↑Wed Feb 05, 2020 5:01 pm
When the digits of 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?
(1) The integer (M - N) has 12 unique factors.
(2) The integer (M - N) is a multiple of 9.
OA
A
Source: Veritas Prep
Say M = 10x + y, where x = tens digit and y = units digit;
Thus, N = 10y + x
Since M > N, we have x > y.
We have to determine the value of M – N = 10 x + y – 10y – x = 9(x – y)
Let's take each statement one by one.
(1) The integer (M - N) has 12 unique factors.
=> 9(x – y) or 3^2*(x – y) has 12 unique factors.
Note that for a number R, which can be written as a^p*b^q, where a and b are prime factors and p and q are positive integers, the number of factors of R is given by (p + 1)*(q + 1).
Again, we know that the number fo factors of M – N = 12 = 3*4 = (2 + 1)*(3 + 1)
Let's rewrite (M – N) as 3^2*(x – y) = 3^2*c^r; where c is prime and r is a positive integer
Thus, the number fo factors of M – N = (2 + 1)*(r + 1)
=> (2 + 1)*(r + 1) = (2 + 1)*(3 + 1) => r = 3
=> x – y = c^3
Since x and y are single digits, with x and y are non-zero integers, the maximum value of x – y is 8. Since c is prime, c can only have one possible value 2; thus, c ^3 = 2^3 = 8.
The value fo M – N = 9(x – y) = 9*2^3 = 9*8 = 72. Sufficient.
(2) The integer (M - N) is a multiple of 9.
We already know this. This is not additional information. Insufficient.
The correct answer:
A
Hope this helps!
-Jay
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