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 N and M?
A) 181
B) 165
C) 121
D) 99
E) 44
[Spoiler Below]
Without loss of generality, le's say M has a as the tens digit, and b as the units digit.
Consequently, N would have b as the tens, and a as the units digit.
remembering how the decimal system works:
M=10⋅a+b
N=10⋅b+a
(adding these two equations)
M+N=11⋅a+11⋅b=11⋅(a+b)
since a and b are both integers, a+b must also be an integer, so M+N=11⋅an∫e≥r
so M + N is a multiple of 11.
A) 181 is prime
B) 165=11⋅5⋅3
C) 121=11⋅11
D) 99=11⋅32
E) 44=11⋅4
(A) is the only choice that isn't a multiple of 11, and therefore cannot be the the sum of M and N.
