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# n = xy35z As shown above, n is a 5-digit even positive integer

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ISB, NUS, NTU Moderator
Joined: 11 Aug 2016
Posts: 311
n = xy35z As shown above, n is a 5-digit even positive integer  [#permalink]

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06 Sep 2018, 11:53
1
8
00:00

Difficulty:

55% (hard)

Question Stats:

66% (02:16) correct 34% (02:46) wrong based on 44 sessions

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n = xy35z
As shown above, n is a 5-digit even positive integer whose ten thousands, thousands and units digits are x, y and z respectively. If all digits of n are distinct and n is divisible by 9 and 55, what is the remainder when the product of x and y is divided by the sum of x and y?

A. 0
B. 1
C. 4
D. 8
E. 12

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Math Expert
Joined: 02 Aug 2009
Posts: 7035
n = xy35z As shown above, n is a 5-digit even positive integer  [#permalink]

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06 Sep 2018, 17:24
1
1
n = xy35z
As shown above, n is a 5-digit even positive integer whose ten thousands, thousands and units digits are x, y and z respectively. If all digits of n are distinct and n is divisible by 9 and 55, what is the remainder when the product of x and y is divided by the sum of x and y?

A. 0
B. 1
C. 4
D. 8
E. 12

Number is
even
Divisible by 9
Divisible by 55 so by 5 and 11
Therefore only possibility is 0 for z
As all are distinct, X and y cannot be 0, 3 or 5

As per divisibility rule of 9,
x+y+3+5+z=x+y+0+8 should be div by 9
Or X+y-1 should be div by 9
all digits are distinct so different possibilities of X and y
1) x+y could be 1, then no possibility
2) x+y could be 10 as 10-1=9, then 9+1 or 8+2 or 6+4

Now for divisibility rule of 11..
x+3-(y+5)=x-y-2 is divisible by 11 or x-y=11a+2
Possibilities
1) x-y=2... then 8-6 or 9-7 or 6-4
2) x-y=11+2=13, then no possibility as X>10

Common amongst two is 6 and 4 so X and y are 6 and 4 in any order..
xy=6*4=24 and x+y=6+4=10
24 divided by 10 will leave 4 as remainder

Ans 4

C
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Joined: 02 Jun 2013
Posts: 21
Re: n = xy35z As shown above, n is a 5-digit even positive integer  [#permalink]

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16 Sep 2018, 06:32
2
xy35z - here x,y,z has to be distinct .

Now since the number is divisible by 55 - 5 and 11. So z has to be 0

now x+y should be equal to 10 . then only it will be divisible by 9

there are three possibilities only with this pair (1,9),(4,6),(2,8)

and when we do xy/(x+y) we get remainder as -9,4,6 respectively.

4 is in option. so its an answer
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Re: n = xy35z As shown above, n is a 5-digit even positive integer &nbs [#permalink] 16 Sep 2018, 06:32
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