problem 2 digit integers, same digits, reverse order

[mepinoargote]

OG 182: 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

M=10t+u and N=10u+t
M+N=11t+11u= 11(t+u)
Must be a multiple of 11, only A is not and thus cannot be the sum of M and N

I don´t understand the official guide explanation, i mean the algebra is pretty simple and makes sense, but how does it come up with the two equations M=10t+u and N= 10 u +t where t and u are 2 digits. Is that a standard rule in this type of questions?

[Patrick_GMATFix]

Any integer can be written as the sum of terms where each term is the product of the digit and its position. For instance, in the number 25, 2 is the tens digit and 5 is the units digit. So 25 = 2*10 + 5*1. Likewise the integer 734 = 7*100 + 3*10 + 4*1.

If the two digit integer is tu where t is the tens and u is the units, then the integer can be written as 10t + u. Another 2-digit integer that has the same digits in reverse order will be 10u + t.


This is OG12 #182. The answer is A. Two detailed solutions are attached. If you cannot see the attachment, read it here

Hope that clears things up!
-Patrick

[mepinoargote]

It helped indeed, thank you so much!

[Patrick_GMATFix]

It helped indeed, thank you so much!

That's what I do

[Asif]

yea. I got that right also. It has to be a multiple of 11. but again don't you think that its on condition where any of the integer cannot be zero.

thanks.

[vipin85]

yea. I got that right also. It has to be a multiple of 11. but again don't you think that its on condition where any of the integer cannot be zero.

thanks.

The question says M and N are positive. And even if one of the digits is zero, their sum would still be a multiple of 11.
eg:

70 and 07 --> Sum = 77
20 and 02 --> Sum = 22

Hope this answers your question.

[Patrick_GMATFix]

70 and 07 --> Sum = 77
20 and 02 --> Sum = 22

Nice, I had not noticed this.

yea. I got that right also. It has to be a multiple of 11. but again don't you think that its on condition where any of the integer cannot be zero.

thanks.

Hi Asif. The integer 05 is a one-digit integer. The leading zero does not count as a digit whenever a GMAT question says an 'n' digit integer. The question says "If the two-digit integers M and N..." In this case, neither one of the integers can be a multiple of 10 (end with zero) because the other integer would not be a 2-digit integer (because the leading zero would not count towards the number of digits).

-Patrick

[Asif]

Thanks Patrick.

[Patrick_GMATFix]

[Tsantsuku]

is it me or all the sums for 2 digit numbers in reverse are multiples of 11? might be a shortcut if this is true!

[Bigred2008]

I got this one right, but I was curious is there ever a situation where you can write a two digit number as 10X-Y rather than 10X+Y?