If the two digit integers M and N are positive and have the

If we have a two digit number AB, where A is the tens digit and B the units digit, then the number is equal to 10A + B.

So here, if A and B represent digits, our numbers are AB and BA. These are equal to 10A + B and 10B + A, and their sum is equal to 11A + 11B. This sum is clearly divisible by 11. So if an answer choice is not divisible by 11, it cannot be our sum, and A, 181, is correct.

AAPL wrote:Official Guide

If the 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

OA A

Since 98 + 89 = 187 and 97 + 79 = 176, we see that there is no way for the sum to be 181.

Alternate solution:

We can let M = 10t + u where t is the tens digit of M and u is the units digit of M. Thus N = 10u + t and M + N = (10t + u) + (10u + t) = 11t + 11u = 11(t + u). We see that the sum of M and N must be a multiple of 11 and all the given choices are multiples of 11 except choice A. So 181 can’t be the sum of M and N.

Answer: A

AAPL wrote:Official Guide

If the 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

OA A

This question relies on our ability to determine the VALUE of any 2-digit number.
For example, what is the VALUE of 83?
For many of us, it has been a very long time since we examined this (we learned this when we were 5 or 6). Most of us just say that 83 has a value of 83, but 83 is really just a "recipe" for determining value.
83 is equal to 8 tens plus 3 ones.
Similarly, 76 is equal to 7 tens plus 6 ones.

In general, if tu represents a 2-digit number (where the t stands for the digit in the tens position, and the u stands for the digit in the units position), then the value of tu is 10t + u

So, if M = tu, then N = ut
The VALUE of M is 10t + u, and the VALUE of N us 10u + t
So, the VALUE of M+N = (10t + u) + (10u + t) = 11t + 11u = 11(t + u)

So, we can see that M+N must be a MULTIPLE OF 11

Answer choices B, C, D and E are all multiples of 11.
Since 181 is not a multiple of 11, M+N cannot equal 181

Answer: A

Cheers,
Brent