Answer:
The value of M is 91.
Explanation:
Suppose T represents the tens digit and U represents the units digit.
Then the numbers M and N are:
M = 10T + U
N = 10U + T
Compute the value of M - N as follows:
M - N = 10T + U - 10U - T = 9T - 9U = 9 (T - U)
Given:
M > N
⇒ T - U > 0
⇒ T > U
Also, M - N is a multiple of 9.
Since M - N is a two-digit number, the possible values of M - N are:
18, 27, 36, 45, 54, 63, and 72
The values 81, 90 and 99 would not be considered since:
9 × 9 = 81
9 × 10 = 90
9 × 11 = 99
T - U = 9 (T - U) = 9 , 10 or 11 respectively and the largest difference for T − U we can get is 9 - 0 = 9 - 0 = 9.
Even this would result in N = 9, which is not a 2-digit number.
It is provided that the integer (M - N) has 12 factors.
Of the possible values of M - N the integer with 12 factors is 72.
The factors of 72 are:
1×72
2×36
3×24
4×18
6×12
8×9
So, M - N = 72.
⇒ M - N = 9 (T - U)
72 = 9 × 8
⇒ T - U = 8
And since T > U, the values of T and U are:
T = 9 and U = 1
We cannot use T - U = 8 - 0 = 8, because then N will be not a two-digit number.
So, M = 91 and N = 19 and their difference is 72.
Thus, the value of M is 91.