When the digits of two-digit, positive integer M are reversed, the result is the two-digit, positive integer N.

[BTGmoderatorDC]

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

[Jay@ManhattanReview]

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
_________________
Manhattan Review

Locations: Manhattan Review Tarnaka | GMAT Prep Madhapur | GRE Prep Kukatpally | Jayanagar GRE Coaching | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.