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

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If the two digit integers M and N are positive and have the same digit  [#permalink]

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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


Question 182 from The Official Guide for GMAT Review 12th Edition:

Originally posted by ellisje22 on 04 Jan 2006, 17:47.
Last edited by Bunuel on 16 Sep 2018, 00:09, edited 4 times in total.
Renamed the topic and edited the question.
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Re: If the two digit integers M and N are positive and have the same digit  [#permalink]

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New post 04 Jan 2006, 18:32
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Let the first digit be xy, then the second digit would be yx.

xy = 10x + y
yx = 10y + x

sum = 11 (x+y)

so the sum has to be multiple of 11. A is not.
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Re: If the two digit integers M and N are positive and have the same digit  [#permalink]

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New post 04 Jan 2006, 19:15
I believe the answer is A and B. Both cannot be expressed as a sum of 2 digit numbers whose digits are the same but interchanged.

Let M= AB(A.. B = 0…9) and N = BA where A and B are the digits 0-9.

- One deducted rule is the last digit of the sum should be the sum of A+B . To illustrate further take for eg 44 the last digit is 4 which can be expressed as a sum of 2+2 , 1+3,0+4. (which means First digit is A and the second B) = 22+22 = 44, Similarly 13+31 = 44, 04+40 = 44. Therefore 44 can be expressed as per above rule. Applying the same rule for 99 = 9 can be expressed as sum of 1+8, 9+0,7+2, 5+4,6+3 all of which can yield 99. Therefore 99 is eliminated. For numbers less than 100 the sum of the last 2 digits cannot exceed 9.
- For numbers greater than 100 the last digit can be expressed as a sum of numbers from 0-9 with one of the numbers > 5. To illustrate this let us analyze 121. Last digit is 1. Since the number is > than 100 we need to express it as a sum of single digit number whose total is 11 therefore it can be 9+2,8+3,7+4,6+5, if we apply any of this we find 92+29 = 121, 83+38 = 121, 74+47+121, 65+56=121. Therefore 121 is eliminated.

164 = 4 can be expressed as 8+6, 9+5,7+7, = 86+68=154,95+59=154,77+77=154. Therefore 164 is one of the choices

Let us look at 181 = 1 can be expressed as 9+2,7+4,8+3,6+5, none of which will yield 181 . Therefore according to me there are 2 answers A or B.

I do not know whether this is the correct way. If someone can elaborate further that will be great.
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Re: Algebra - Applied Problems  [#permalink]

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New post 18 Sep 2013, 18:16
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simple solution:
M=10x+y
N=10y+x
M+N=11x+11y=11(x+y)
In other words the answer is a multiple of 11.
Now the question becomes " which of the following is NOT a multiple of 11?"
answer----->181----->choice A
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Re: Algebra - Applied Problems  [#permalink]

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New post 16 Mar 2014, 18:17
madn800 wrote:
simple solution:
M=10x+y
N=10y+x
M+N=11x+11y=11(x+y)
In other words the answer is a multiple of 11.
Now the question becomes " which of the following is NOT a multiple of 11?"
answer----->181----->choice A


madn800 is the best solution in my opinion.. :D

But, just to give another solution I've just saw: think of M = AB and N = BA.
(1) If A+B < 10, then:

AB
BA +
-----
B B ---> sum will result in a two digit number
+ +
A A

The sum of A + B will be less than 10, so the summed number digit's will be the same, so D and E satisfy. Now... if
2) A+B > 10, then:

AB
BA
----
1 B ---> sum will result in a three digit number
+ +
B A
+
A

Since B+A in this case is greater than 10, you should add a plus 1 in the sum of the next digit, then, this next digit will be the first digit + 1. In a 3 digit number __ ___ ___, the first two digits will be __ A+B+1 A+B , so B and C satisfy, leaving A as the only choice left.
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Re: If the two-digit integers M and N are positive and have the  [#permalink]

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New post 19 Apr 2015, 17:48
Hi All,

Since the question asks for the answer that CANNOT be the sum of M and N, and the answers are numbers, we can use a combination of TESTing VALUES and TESTing THE ANSWERS to eliminate the possible values and find the answer to the question.

We're told that M and N are two-digit positive integers and have the SAME DIGITS but in REVERSE ORDER. We're asked which of the 5 answers CANNOT be the SUM of M and N.

Let's start with the 'easiest' answer first:

44. Can we get to 44 in the manner described?
Yes, if the numbers are 13 and 31.....13+31 = 44. Eliminate Answer E

Now let's work through the rest of the list....

Can we get to 99 in the manner described?
Yes, there are several ways to do it. For example, if the numbers are 18 and 81.....18+81 = 99. Eliminate Answer D

Can we get to 121 in the manner described?
Yes, there are several ways to do it. For example, if the numbers are 38 and 83.....38+83 = 121. Eliminate Answer C

Can we get to 165 in the manner described?
Yes, there are a couple of ways to do it. For example, if the numbers are 78 and 87.....78+87 = 165. Eliminate Answer B

There's only one answer left....

Final Answer:

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Re: If the two digit integers M and N are positive and have the same digit  [#permalink]

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New post 18 Aug 2015, 21:21
1
Hi All,

This type of question can be solved with algebra or "brute force"; here's how you use "brute force" to figure out what's possible and what's not.

We're told that M and N are each two-digit numbers with the two digits in reverse order (e.g. 23 and 32). We're asked which of the 5 answers CANNOT BE the sum of the M and N.

Let's see if we can sum to the given answers (starting with the easiest; keep in mind that there might be more than one way to "hit" each answer):

44? 13 + 31
99? 45 + 54
121? 56 + 65
165? 78 + 87

Since we can sum to each of those 4 answers, there's only one answer left...

Final Answer:

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Re: If the two digit integers M and N are positive and have the same digit  [#permalink]

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New post 26 Aug 2015, 00:47
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ellisje22 wrote:
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


Remember this as a formula/theorem. This helps me solve all the \(m & n\) \(2-digit\) \(reversed\) \(problems\)

If M is a 2 digit number and n is obtained by reversing the digits of m, then:

1. The sum is a multiple of 11
2. Difference is a multiple of 9


Back to the question. Since the question asks us about the sum, the ans would be the option that will not be divisible by 11

Ans: A

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Re: If the two digit integers M and N are positive and have the same digit  [#permalink]

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New post 15 Sep 2018, 13:07
Bunuel

Same question as: https://gmatclub.com/forum/if-the-two-d ... 87908.html
Probably retire?
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Re: If the two digit integers M and N are positive and have the same digit  [#permalink]

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New post 16 Sep 2018, 00:10
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Re: If the two digit integers M and N are positive and have the same digit &nbs [#permalink] 16 Sep 2018, 00:10
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