Data Sufficiency

BTGmoderatorDC wrote: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; thus, N = 10y + x, where x > y

Let's take each statement one by one.

(1) The integer (M - N) has 12 unique factors.

M - N = 10x + y - 10y - x = 9(x - y) = 3^2*(x - y)

So, 3^2*(x - y) have 12 unique factors.

Note that for a any number say P = a^m*b^n, there are (m + 1)*(n + 1) distinct factors, where a and b are prime numbers.

Coming back to 3^2*(x - y) have 12 unique factors, say r^s = x - y. Thus, 3^2*(x - y) = 3^2*r^s have 12 unique factors.

=> (2 + 1)*(s + 1) = 12 => s = 3

So, x - y = r^3

Note that x and y are single non-zero digits and x > y, the maximum value of x - y would be 9 - 1 = 8. Thus, r^3 ≤ 8. r cannot be 1 since r =1, we'll have x = y, not possible; thus, r = 2 and r^3 = 8 = x - y. This gives x = 9 and y = 1.

Thus, M = 91. Sufficient.

(2) The integer (M - N) is a multiple of 9.

M - N = 10x + y - 10y - x = 9(x - y)

Since x - y may have many values, we cannot find out the unique value of x and y; thus, the unique value of M is not possible. Insufficient.