Problem Solving

If two-digit integers M and N are positive and have the same digits, but in reverse order, which of the following CANNOT be the sum of M and N?

A. 181
B. 165
C. 121
D. 99
E. 44

Source : OG 12th Edition, Q-182
Official Ans. A

Is there any shortcut/efficient method ?

sodha.rakesh wrote:If two-digit integers M and N are positive and have the same digits, but in reverse order, which of the following CANNOT be the sum of M and N?

A. 181
B. 165
C. 121
D. 99
E. 44

Source : OG 12th Edition, Q-182
Official Ans. A

Is there any shortcut/efficient method ?

The number which is not divisible by 11 will be the ans.
say two digits are a & b.
So number M = 10a+b
Number N = 10b + a
M + N = (10a + b ) + (10b +a) = 11a + 11b = 11(a+B) .. so to be the sum of M and N is has to be divisible by 11.

sodha.rakesh wrote:If two-digit integers M and N are positive and have the same digits, but in reverse order, which of the following CANNOT be the sum of M and N?

A. 181
B. 165
C. 121
D. 99
E. 44

Source : OG 12th Edition, Q-182
Official Ans. A

Is there any shortcut/efficient method ?

Let the numbers be 10x+y (x is the ten's digit, and y the unit's) and 10y + x (y the ten's digit, and x the unit's)

sum = 10x+y + 10y+x = 11(x+y)

Thus, 11 should be a factor of the number.

Answer choice A is the only one that is not divisible by 11, and thus the correct answer.

sodha.rakesh wrote:If two-digit integers M and N are positive and have the same digits, but in reverse order, which of the following CANNOT be the sum of M and N?

A. 181
B. 165
C. 121
D. 99
E. 44

Source : OG 12th Edition, Q-182
Official Ans. A

Is there any shortcut/efficient method ?

Look at the units digits of the answer choices.
For the sum of M and N to have a units digit of 1, the sum of the units digits of M and the units digit of N must be 11.
Thus, to obtain answer choice A (181) or answer choice C (121), the units digits of M and N must be either 2 and 9, 3 and 8, 4 and 7, or 5 and 6.

29 + 92 = 38 + 83 = 47 + 74 = 56 + 65 = 121.

No way to get a sum of 181.

The correct answer is A.

Thanks Mitch for such nice and fast approches , normally i do get involved into the making equations and solving the questions by equations whereas this method is pretty fast and simple . May be i cant get rid of my Engineering b/g effect .

GMATGuruNY wrote:

sodha.rakesh wrote:If two-digit integers M and N are positive and have the same digits, but in reverse order, which of the following CANNOT be the sum of M and N?

A. 181
B. 165
C. 121
D. 99
E. 44

Source : OG 12th Edition, Q-182
Official Ans. A

Is there any shortcut/efficient method ?

Look at the units digits of the answer choices.
For the sum of M and N to have a units digit of 1, the sum of the units digits of M and the units digit of N must be 11.
Thus, to obtain answer choice A (181) or answer choice C (121), the units digits of M and N must be either 2 and 9, 3 and 8, 4 and 7, or 5 and 6.

29 + 92 = 38 + 83 = 47 + 74 = 56 + 65 = 121.

No way to get a sum of 181.

The correct answer is A.

Thank you for your reply and good strategy (that is the reason why I posted this que. here), Would you please provide some details why we are considering the answer choices 121 & 181 only and how to eliminate other options e.g. 99,44,165?

sodha.rakesh wrote:

GMATGuruNY wrote:

sodha.rakesh wrote:If two-digit integers M and N are positive and have the same digits, but in reverse order, which of the following CANNOT be the sum of M and N?

A. 181
B. 165
C. 121
D. 99
E. 44

Source : OG 12th Edition, Q-182
Official Ans. A

Is there any shortcut/efficient method ?

Look at the units digits of the answer choices.
For the sum of M and N to have a units digit of 1, the sum of the units digits of M and the units digit of N must be 11.
Thus, to obtain answer choice A (181) or answer choice C (121), the units digits of M and N must be either 2 and 9, 3 and 8, 4 and 7, or 5 and 6.

29 + 92 = 38 + 83 = 47 + 74 = 56 + 65 = 121.

No way to get a sum of 181.

The correct answer is A.

Thank you for your reply and good strategy (that is the reason why I posted this que. here), Would you please provide some details why we are considering the answer choices 121 & 181 only and how to eliminate other options e.g. 99,44,165?

Two reasons I focused on answer choices A and C:

-- they each have the same units digit (1), increasing the likelihood that one of them won't be possible.
-- there is only way to get a units digit of 1: the sum of the digits M and N must be 11.

Once I determined that MN + NM = 181 is impossible, there was no reason to check the other answer choices.

The other answers choices can be eliminated because they are all easily achieved:

E) 44 = 13+31 (any two digits whose sum is 4 will work)
D) 99 = 45+54 (any two digits whose sum is 9 will work)
B) 165 = 78+87 (any two digits whose sum is 15 will work)

Hope this helps!