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Math Expert V
Joined: 02 Sep 2009
Posts: 59588
The following addition operation shows the sum of the two-digit positi  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 71% (02:11) correct 29% (02:03) wrong based on 127 sessions

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The following addition operation shows the sum of the two-digit positive integers XY and YX. If the three-digit integer XXZ, and X, Y , and Z are different digits, what is the value of the integer Z?

XY
+YX
_____
XXZ

(A) 8
(B) 7
(C) 2
(D) 1
(E) 0

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Director  G
Joined: 20 Feb 2015
Posts: 737
Concentration: Strategy, General Management
The following addition operation shows the sum of the two-digit positi  [#permalink]

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Bunuel wrote:
The following addition operation shows the sum of the two-digit positive integers XY and YX. If the three-digit integer XXZ, and X, Y , and Z are different digits, what is the value of the integer Z?

XY
+YX
_____
XXZ

(A) 8
(B) 7
(C) 2
(D) 1
(E) 0

Given the options , we want the units digit of XXZ
consider 10
10 can be written as 19, 91
19+91= 110

Therefore E
GMATH Teacher P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Re: The following addition operation shows the sum of the two-digit positi  [#permalink]

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Bunuel wrote:
The following addition operation shows the sum of the two-digit positive integers XY and YX. If the three-digit integer XXZ, and X, Y , and Z are different digits, what is the value of the integer Z?

XY
+YX
_____
XXZ

(A) 8
(B) 7
(C) 2
(D) 1
(E) 0

$$X,Y\,\,\, \in \,\,\,\,\left\{ {1,2, \ldots ,9} \right\}$$
$$Z \in \,\,\,\,\left\{ {0,1,2, \ldots ,9} \right\}$$
$$X,Y,Z\,\,\,{\text{distinct}}$$

$$? = Z$$

$$\left\langle {XXZ} \right\rangle = \left\langle {XY} \right\rangle + \left\langle {YX} \right\rangle < 99 + 99 < 200\,\,\,\,\,\,\, \Rightarrow \,\,\,\,X = 1$$

$$\left\langle {1Y} \right\rangle + \left\langle {Y1} \right\rangle = \left\langle {11Z} \right\rangle \,\,\,\,\, \Rightarrow \,\,\,Y = 9\,\,\,\,\,\,\left( {18 + 81 < 100} \right)$$

$$\left\langle {11Z} \right\rangle = 19 + 91 = 110\,\,\,\,\, \Rightarrow \,\,\,Z = 0\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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The following addition operation shows the sum of the two-digit positi  [#permalink]

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Bunuel wrote:
The following addition operation shows the sum of the two-digit positive integers XY and YX. If the three-digit integer XXZ, and X, Y , and Z are different digits, what is the value of the integer Z?

XY
+YX
_____
XXZ

(A) 8
(B) 7
(C) 2
(D) 1
(E) 0

XXZ has to be a multiple of 11
with a 100s digit of 1
and a matching tens digit
the only possibility is 110
0
E
Manager  S
Joined: 16 Mar 2017
Posts: 61
Location: France
Concentration: Marketing, Strategy
GMAT 1: 640 Q38 V40 GMAT 2: 710 Q47 V41 WE: Marketing (Retail)
Re: The following addition operation shows the sum of the two-digit positi  [#permalink]

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Considering the stem, the sum of two-digit integer must result in a three-digit integer with the first 2 digits of this integer being similar.
The only case possible is for X to equal one (for example no two digit integer can sum above 198), hence we now have 1Y+Y1 = 11Z
Which leave us with the only option possible for Y to yield a three digit number : 9 (for example 8 for Y would yield : 18+81= 99)
Finally : 19+91=110
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Intern  B
Joined: 22 Jun 2017
Posts: 9
Re: The following addition operation shows the sum of the two-digit positi  [#permalink]

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fskilnik wrote:
Bunuel wrote:
The following addition operation shows the sum of the two-digit positive integers XY and YX. If the three-digit integer XXZ, and X, Y , and Z are different digits, what is the value of the integer Z?

XY
+YX
_____
XXZ

(A) 8
(B) 7
(C) 2
(D) 1
(E) 0

$$X,Y\,\,\, \in \,\,\,\,\left\{ {1,2, \ldots ,9} \right\}$$
$$Z \in \,\,\,\,\left\{ {0,1,2, \ldots ,9} \right\}$$
$$X,Y,Z\,\,\,{\text{distinct}}$$

$$? = Z$$

$$\left\langle {XXZ} \right\rangle = \left\langle {XY} \right\rangle + \left\langle {YX} \right\rangle < 99 + 99 < 200\,\,\,\,\,\,\, \Rightarrow \,\,\,\,X = 1$$

$$\left\langle {1Y} \right\rangle + \left\langle {Y1} \right\rangle = \left\langle {11Z} \right\rangle \,\,\,\,\, \Rightarrow \,\,\,Y = 9\,\,\,\,\,\,\left( {18 + 81 < 100} \right)$$

$$\left\langle {11Z} \right\rangle = 19 + 91 = 110\,\,\,\,\, \Rightarrow \,\,\,Z = 0\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Can you explain how u got the below plzz?

⟨XXZ⟩=⟨XY⟩+⟨YX⟩<99+99<200⇒X=1⟨XXZ⟩=⟨XY⟩+⟨YX⟩<99+99<200⇒X=1???

⟨1Y⟩+⟨Y1⟩=⟨11Z⟩⇒Y=9???(18+81<100)
GMATH Teacher P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Re: The following addition operation shows the sum of the two-digit positi  [#permalink]

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2
ias882000 wrote:
fskilnik wrote:
Bunuel wrote:
The following addition operation shows the sum of the two-digit positive integers XY and YX. If the three-digit integer XXZ, and X, Y , and Z are different digits, what is the value of the integer Z?

XY
+YX
_____
XXZ

(A) 8
(B) 7
(C) 2
(D) 1
(E) 0

$$X,Y\,\,\, \in \,\,\,\,\left\{ {1,2, \ldots ,9} \right\}$$
$$Z \in \,\,\,\,\left\{ {0,1,2, \ldots ,9} \right\}$$
$$X,Y,Z\,\,\,{\text{distinct}}$$

$$? = Z$$

$$\left\langle {XXZ} \right\rangle = \left\langle {XY} \right\rangle + \left\langle {YX} \right\rangle < 99 + 99 < 200\,\,\,\,\,\,\, \Rightarrow \,\,\,\,X = 1$$

$$\left\langle {1Y} \right\rangle + \left\langle {Y1} \right\rangle = \left\langle {11Z} \right\rangle \,\,\,\,\, \Rightarrow \,\,\,Y = 9\,\,\,\,\,\,\left( {18 + 81 < 100} \right)$$

$$\left\langle {11Z} \right\rangle = 19 + 91 = 110\,\,\,\,\, \Rightarrow \,\,\,Z = 0\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Can you explain how u got the below plzz?

⟨XXZ⟩=⟨XY⟩+⟨YX⟩<99+99<200⇒X=1???

⟨1Y⟩+⟨Y1⟩=⟨11Z⟩⇒Y=9???(18+81<100)

Hi, ias882000 !

Thank you for your interest in my solution.

1st doubt: <XXZ> is a positive 3-digit number less than 200, therefore X must be 1.

2nd doubt: If Y is 8 or less, ⟨1Y⟩+⟨Y1⟩ would be 18+81 or less, hence it would not be a three-digit number.
It must be, to be equal to ⟨11Z⟩.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Director  V
Joined: 27 May 2012
Posts: 945
Re: The following addition operation shows the sum of the two-digit positi  [#permalink]

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Bunuel wrote:
The following addition operation shows the sum of the two-digit positive integers XY and YX. If the three-digit integer XXZ, and X, Y , and Z are different digits, what is the value of the integer Z?

XY
+YX
_____
XXZ

(A) 8
(B) 7
(C) 2
(D) 1
(E) 0

Dear Moderator,
This same problem has been duplicated in the link below, you may wish to merge the same, Thank you.

https://gmatclub.com/forum/in-the-corre ... ml#p384737
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- Stne Re: The following addition operation shows the sum of the two-digit positi   [#permalink] 19 Dec 2018, 13:21
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