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
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When the digits of two-digit, positive integer M are reverse
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RBBmba@2014
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RBBmba@2014 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.
Let T = the TENS digit of M and U = the nonzero UNITS digit of M.gmat_winter 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.
Then:
M = 10T + U.
Since the digits in N are reversed, N = 10U + T.
Resulting difference:
M-N = (10T + U) - (10U + T) = 9T - 9U = 9(T-U).
Since the greatest possible value of T-U = 9-1 = 8, M-N must be a MULTIPLE OF 9 NO GREATER THAN 9*8 = 72.
Thus, M-N = 9(T-U) must be equal to one of the following values:
9, 18, 27, 36, 45, 54, 63, 72.
Statement 1: The integer (M - N) has 12 unique factors.
Of the options for M-N, only 72 has 12 unique factors:
1*72
2*36
3*24
4*18
6*12
8*9.
Total factors = 12.
Since M-N = 9(T-U) = 72, we get:
9(T-U) = 72
T-U = 8, implying that T=9 and U=1.
Result:
M = 10T + U = 10(9) + 1 = 91.
SUFFICIENT.
Statement 2: The integer (M - N) is a multiple of 9.
No new information.
The prompt on its own implies that M-N is a multiple of 9.
INSUFFICIENT.
The correct answer is A.
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As per the given info
M= 10a + b
N= 10b + a
M> N
Statement 1: The interger (M-N) has 12 unique factors
M-N = (10a+ b) - (10b + a)
M-N = 9a - 9b
M- N = 9(a-b)
$$M\ -\ N\ =\ 3^2\left(a-b\right)$$
Statement 1 tells us that M - N has 12 factors, this means that (a-b) should be a number with power as 3
$$a-b\ =\ x^3$$
if x = 2
$$a-b\ =\ 2^3$$
a-b = 8
i.e a=9 and b = 1
9-1 = 8
so M= 91
N= 19 (given that M>N, Statement 1 is sufficient)
Statement 2: The integer (M-N) is a multiple of 9
M-N = (10a + b) - (10b + a) = 9a - 9b.
M-N = 9(a-b)
This is already a multiple of 9, therefore (a-b) can be any integer. Hence we cannot narrow down the values of a and b to find M.
Statement 2 is then termed as insufficient.
Answers = A because statement 1 above is sufficient, but statement 2 alone is not sufficient.
M= 10a + b
N= 10b + a
M> N
Statement 1: The interger (M-N) has 12 unique factors
M-N = (10a+ b) - (10b + a)
M-N = 9a - 9b
M- N = 9(a-b)
$$M\ -\ N\ =\ 3^2\left(a-b\right)$$
Statement 1 tells us that M - N has 12 factors, this means that (a-b) should be a number with power as 3
$$a-b\ =\ x^3$$
if x = 2
$$a-b\ =\ 2^3$$
a-b = 8
i.e a=9 and b = 1
9-1 = 8
so M= 91
N= 19 (given that M>N, Statement 1 is sufficient)
Statement 2: The integer (M-N) is a multiple of 9
M-N = (10a + b) - (10b + a) = 9a - 9b.
M-N = 9(a-b)
This is already a multiple of 9, therefore (a-b) can be any integer. Hence we cannot narrow down the values of a and b to find M.
Statement 2 is then termed as insufficient.
Answers = A because statement 1 above is sufficient, but statement 2 alone is not sufficient.
