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Jamboree and GMAT Club Contest: The sum of the two digit numbers

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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 08 Nov 2015, 11:01
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X, Y, P & Q all are unique digit. Best and least time way is to try the options. Option III) - If X = 9, then Y must be '6', so that Y+X = 15. 1 carry over and so X+Y = 16, i.e, Sum = 165; i.e., P=1, Q=6, X=9, & Y=6. Since here Q=Y=6, whereas question state that all are different, this is not correct.
Now option II): X=8, that means Y = 7 and hence sum will be 165; with P=1, Q=6, X=8 & Y=7. Since all are unique, this can be the answer.
Similarly option I): X=7 will give Y=8, P=1, Q=6, which also is true with the given condition.

Hence, correct answer is - D :roll:
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 08 Nov 2015, 12:12
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The Sum is PQ5, and it is written that P, Q, X and Y are different.
So what CAN be the value of X?
Going by the option given X=7 so putting it in given condition, we get X=7 Y=8 and P=1 Q=6.-- so 1st is possible.
Going by the option given X=8 so putting it in given condition, we get X=8 Y=7 and P=1 Q=6.-- so 2nd is possible.
Going by the options given X=9 so putting it in given condition, we get X=9 Y=6 and P=1 Q=6.-- so 3rd is not possible as Y and Q are same and this condition is not allowed.
So Answer is D
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 08 Nov 2015, 18:17
The possible values XY can take are: 96, 69, 87, 78. (23 and 32 are eliminated as their sum gives two digit number.)
Upon adding reverse of each of the above number with the corresponding original number, one will get a number in 'PQ5' form.

Thus, the different values X can take are: 6, 7, 8 and 9.

IMO : E.

Please correct me if wrong.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 08 Nov 2015, 18:44
The only combinations for which the units digit (Y+X) equals to 5 are (6,9)(9,6) (7,8) (8,7) fora three digit number. Hence all the options given are possible values.

Option E.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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Since XY + YX represents a three digit number, X+ Y must be greater than 10.
Now, the last digit is 5 so there are only two possibilities left for X and Y : (7,8) or (9,6). Both of these possibilities add up to give 15.
Given that X, Y , P and Q represent different integers, we cannot have (9,6) as our value for X or Y because
96 + 69 = 165 => X= Q => this is not possible.
So the only alternative left is for X and Y to be either (7,8)
So , X can take two values 7 or 8
Ans.: D
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 11 Nov 2015, 06:51
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Given:

XY
+YX
-------
PQ5
------
X,Y,P & Q Different non zero number.

- Hence X,Y,P,&Q cant be 5 and also X + Y or Y + X = gives the unit digit as 5.

When ever answering these sort of questions its always good to start from the numbers present in the answer choice. This will save lot of time.
The given numbers are 7,8 & 9

Option 1 : If Y = 7, then X = 8 hence we will have 5 in the unit digit.
So 78 + 87 = 165 and all the numbers are unique. Hence this is correct.

Option 2 : Since Option 1 and Option 2 are interchangeable we are not performing the same thing again and Option 2 is also correct.

Option 3: 9 + 6 = Gives 5 in the unit digit. Hence 96 + 69 = 165 which is not correct since X = Q. 6 repeats for both X and Q. But according to the question it should not since all are different numbers.

Hence the correct answer choise is (D). Which is Option 1 and Option 2 are the correct answers.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 11 Nov 2015, 07:50
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go by options

If x=7, y must be 8

7 8
+8 7
________
1 6 5 ......... P,Q,X,Y,all are different numbers

If x=8, y must be 7

8 7
+7 8
________
1 6 5 ......... P,Q,X,Y are different

If x=9, y must be 6

9 6
+6 9
________
1 6 5 ......... here Y=Q condition fails.

so answer is option "D"
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 11 Nov 2015, 11:03
X Y
+Y X
________
P Q 5

In-order to give rise to 3 digit no when added we need to select set of integers will result in 2 digit when added and 5 in units place.

X/Y can be {7,8} or {6,9}

out if all these data, value of X can be any one of the four values above. {7/8/6/9}

So answer is (E) I , II and III
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 12 Nov 2015, 11:18
I just plugged in 69+96 , 87+78 and 78+87 (all three work)
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 12 Nov 2015, 11:25
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Answer is D because when we add XY=87 and YX=78 (or vice versa) we get PQ5=165 where X=8 OR X=7. Also we can see here X,Y,P,Q all are different.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 12 Nov 2015, 13:32
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XY
+YX
---------
PQ5

Now, we know that P,Q,X,Y are distinct non-zero digits. Again, (XY + YX) will always be divisible by 11.....(a)
{
Since, XY = 10X + Y and YX = 10Y + X
Therefore, XY + YX = 11(X+Y)
}
Now from (a) and by the question PQ5 is a there digit number, the only possible value of PQ5 which is divisible by 11 and having unit digit 5 is 165.
Therefore, P=1, Q=6.....(b)
Again, PQ5 = XY + YX = 11(X+Y)......from (a)
or, 11(X+Y) = 165
or, X+Y = 15.........(c)

Now since X,and Y are single digit numbers, therefore
either {X,Y} = {7,8} or {8,7}
Or
{X,Y} ={9,6} or {6,9}

78(XY)
+87(YX)
---------
165(PQ5)

Or

69(XY)
+96(YX)
-----------
165(PQ5)

But from the question stem, P,Q,X,Y are distinct non-zero digits. Therefore {X,Y} = {6,9} or {9,6} is not possible., since in this case P = X = 6.

Therefore the possible values of X is 7, or 8.

The the option id "D", I & II Only.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 12 Nov 2015, 16:28
Answer d number 7&8
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 12 Nov 2015, 21:30
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X Y
+Y X
________

The sum of the two digit numbers above is a three digit number PQ5, where each letter X, Y, P, and Q represents a different non zero digit. Which of the following can be the value of X?

I) 7
II) 8
III) 9


Method used Plugging in.

If x=7, Y should be 8(to get unit digit of sum as 5)
so,
78
87
---
165

x=7, y=8, p=1, q=6. Satisfies the condition that all digits are different. So x can be 7

If x=8, Y should be 7(to get unit digit of sum as 5)
so,
87
78
---
165

x=8, y=7, p=1, q=6. Satisfies the condition that all digits are different. So x can be 8


If x=9, Y should be 6(to get unit digit of sum as 5)
so,
96
69
---
165

x=9, y=6, p=1, q=6. Does not satisfies the condition that all digits are different. y=q=6. So x cannot be 9

Ans: (D) I and II only
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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Using Back-solving method to answer the question.
We have to choose X which when add with Y comes upto be "15" because we have unit digit available as 5 (in the Question QP5).
So, choosing X=8, then Y must be 7, if you put these values above, you will get the answer 165.
Now, choose X= 7, then Y must be 8, so putting these will have same answer too as 165
Now if we choose X=9, then Y must be 6 to make unit digit 5, but this way Y & P will have same value as 6 but question states that X,Y,Q,P are all different non zero numbers, So X can be I, II only. Option D
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Originally posted by abdulraufliaqat on 12 Nov 2015, 23:07.
Last edited by abdulraufliaqat on 13 Nov 2015, 04:03, edited 1 time in total.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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D ie 1&2 only
X &Y can't be less than 5 else ans would have been 55 not PQ5
So options are 7,8 or 6,9
Manually add 69+96 = 165 thus Q=x or y
So not possible
Next 78+87=165 which provides all unique nos
Thus X=7 or 8
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 13 Nov 2015, 00:13
XY
+YX
-------
PQ5

Logically, there are 2 types of addition resulting in 5 in the unit digit: 5 and 15.

If X+Y=5 then X and Y must each be less than 5 (since 0<X, Y<9). This means the result can only be a 2-digit number => eliminate this possibility.
If X+Y= 15 we have 4 sets: {X,Y}={(7,8); (8,7); (6,9); (9,6)}. All these sets have result of 165, satisfy conditions that X, Y, P, Q are different non-zero digits. So X can be 6, 7, 8, 9.

Answer is E.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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X Y
Y X
-----
P Q 5
------
Going by answer choices , X can be either 7,8 or 9
1)First lets substitute X =7
if X =7 , Y should be equal to 8 to get the last digit as 8. So the number will be
78
+87
-----
165
----

2)Now lets substitute, X=8 and therefore Y becomes 7 to get last digit as 5.

87
78
---
165
-----

3)Now lets substitute, X=9 and now Y becomes 6 to get last digit as 5.
96
+69
----
165
----
But since both Y=Q=6, therefore X cannot be equal to 9. So X can be equal to only 7 or 8 which is choice D.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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Explanation to Qn#1:
XY can be written as 10X+Y
YX can be written as 10Y+X
Adding both gives 11X + 11Y or 11(X + Y)
Given, 11(X + Y) = PQ5, implies that X + Y has to end with a 5, so that its product with 11 yields a number ending in a 5

Now, out of the given choices, both 7 & 8 satisfy the above condition for X.
As far as 9 is concerned, if it were true, then since X is 9, then Y must be 6 to yield a sum ending with 5. But in that case, 9 + 6 shall yield a carry forward of 1, which, when added to the overall sum, will result in a digit repeated (thus contradicting the given condition of having different digits for X, Y, P and Q.

Hence only possible solutions are 7 and 8. Thus answer is D i.e. I and II only.

We can verify by plugging our answer choices -> 78 + 87 = 165 (satisfied) and 69 + 96 = 165 (not satisfied bcoz Q and X/Y a not distinct).
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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New post 13 Nov 2015, 08:42
X Y
+Y X
________

The sum of the two digit numbers above is a three digit number PQ5, where each letter X, Y, P, and Q represents a different non zero digit. Which of the following can be the value of X?

I) 7
II) 8
III) 9

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I , II and III
Explanation:-
X Y
+Y X
________
PQ5
Above summation implies X+Y=15 (5 is not possible as P is not zero,Given).This implies XY could be69, 96,87 or 78.Here PQ5 =165 Hence Q=6.Therefore X could be 7,8 or 9( not 6 as P,Q,X,Y are all different non zero digit and Q=6).
Therefore Correct answer is "E".
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers [#permalink]

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The answer is D.

According to the question, the last digit of the result from addition is 5.
Thus, X + Y = 15, since X + Y can't be 5 because both the digits would then yield 55(two digits) as a result. but the question states that the result must be 3 digits i.e. PQ5.

So it also means that the sum must be 165, regardless of the value of X and Y.
Here, X and Y must be 7 and 8, or 6 and 9 (not respectively) since X + Y = 15.
Now, the question says that P,Q,X and Y are different integers.
So, X and Y can't be 6 and 9 since Q=6. Thus XY must be either 78 or 87.
Therefore, X can be either 8 or 7. So the correct answer is D.
Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers   [#permalink] 13 Nov 2015, 08:55

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