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Is a three-digit number xyz divisible by 7?

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Is a three-digit number xyz divisible by 7? [#permalink]  21 Aug 2014, 00:42
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Question Stats:

39% (02:09) correct 61% (01:22) wrong based on 41 sessions
Is a three-digit number xyz divisible by 7?

(1) |2*z - 10x - y| is divisible by 7
(2) (z + 3y + 2x) is divisible by 7
[Reveal] Spoiler: OA

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Re: Is a three-digit number xyz divisible by 7? [#permalink]  21 Aug 2014, 06:28
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Good Question!
Given 3 digit number is xyz and can be written as 100*x + 10*y + z
{ Ex: 123 can be written as 100*1 + 10*20 + 1*3 = 123 }

STAT1
|2*z -10x -y| is divisible by 7
=> |2*z -10x - y| = 7k (where k is an integer)
=> 2z - 10x -y = 7r (where r can be positive or negative depending on the sign of 2z - 10x - y (doesn't really matter in this question till the time we know that its a multiple of 7))

multiply both sides by -10 we get

-20z + 100x + 10y = -70r
-21z + z + 100x + 10y = -70r (breaking -20z into -21z + z)
or, 100x + 10y + z = 21z - 70r
21z - 70r is a multiple of 7
=> 100x + 10y + z is a multiple of 7, hence divisible by 7
So, xyz is divisible by 7
So, SUFFICIENT

STAT2
z + 3y + 2x = 7k (where k is an integer)

multiply both sides by 50 we get
50z + 150y + 100x = 350k

49z + z + 140y+ 10y+ 100x = 350k
or, 100x + 10y + z = 350k - 140y - 49z

350k - 140y - 49z is divisible by 7
So, 100x + 10y + z is also divisible by 7
So, xyz is divisible by 7
So, SUFFICIENT

Hope it helps!

Is a three-digit number xyz divisible by 7?

(1) |2*z - 10x - y| is divisible by 7
(2) (z + 3y + 2x) is divisible by 7

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Re: Is a three-digit number xyz divisible by 7? [#permalink]  21 Aug 2014, 06:35
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Is a three-digit number xyz divisible by 7?

(1) |2*z - 10x - y| is divisible by 7
(2) (z + 3y + 2x) is divisible by 7

Nice Problem, Looks difficult but easy one.

First we need to check Is xyz is divisible by 7.
write 3 digit number as xyz= 100x+10y+z , You need to check is this number divisible.

Using Statement 1) (2z-10x-y) is divisible by 7
so it can be re written as 2z-10x-y = 7*N (where N is the interger which multiply 7 gives the result.)
10x = 2z-y-7n

putting the value of 10x in 100x+10y+z
10*10x+10y+z
10(2z-y-7n)+10y+z
20z-10y-70n+10y+z
21z-70N
7(3z-10N) So this is multiple of 7.

Using Statement 2) z + 3y + 2x = 7*K(where N is the interger which multiply 7 gives the result)
z=7k-3y-2x
putting this value of z in 100x+10y+z
100x+10y-7k-3y-2x
98x-7k+7y
7(14x-k+y)
which is also divisible by 7.

Hence D is the right answer.
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Re: Is a three-digit number xyz divisible by 7? [#permalink]  21 Aug 2014, 15:21
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Is a three-digit number xyz divisible by 7?

(1) |2*z - 10x - y| is divisible by 7
(2) (z + 3y + 2x) is divisible by 7

(1) |2*z - 10x - y| is divisible by 7
Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7
(including 0), then the number is divisible by 7
--> (1) is sufficient

(2) (z + 3y + 2x) is divisible by 7
It is one of divisibility rules for 7

To know if a number is a multiple of seven or not, we can use also 3 coefficients (1 , 2 , 3). We multiply the first number starting from the ones place by 1,
then the second from the right by 3, the third by 2, the fourth by -1, the fifth by -3, the sixth by -2, and the seventh by 1, and so forth.

Example: 348967129356876.

6 + 21 + 16 - 6 - 15 - 6 + 9 + 6 + 2 - 7 - 18 - 18 + 8 + 12 + 6 = 16
means the number is not multiple of seven.

If the number was 348967129356874, then the number is a multiple of seven because instead of 16, we would find 14 as a result, which is a multiple of 7

--> (2) is sufficient

http://mathforum.org/k12/mathtips/division.tips.html
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Re: Is a three-digit number xyz divisible by 7? [#permalink]  21 Aug 2014, 19:30
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Is a three-digit number xyz divisible by 7?

(1) |2*z - 10x - y| is divisible by 7
(2) (z + 3y + 2x) is divisible by 7

My Approach:

Since xyz is 3 digit number,lets select some nos and see what the expression ins St1 and St2 give us

Consider, xyz=105,112,126 (All Multiples of 7)
and 108,117,131
xyz=105
St1 |2*5-10*1-0|=0-----> is divisble by 7

xyz=119
St 1 |2*9-10*1-1|=|7|-----> Divisble by 7

Consider x=108
St 1|2*8-10*1-0| =|6|----> Not divisble by 7

St1 is sufficient. Option B,C and E cancelled out

St2 (z + 3y + 2x)---> We use the same examples and see that xyz is divisble by 7

105----> 5+3*0+2*1)=7---Divisbly by 7

108-----> 8+3*0+2*1=10----> Not divisbly by 7..

Ans is D
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Re: Is a three-digit number xyz divisible by 7? [#permalink]  27 Aug 2014, 16:23
Is a three-digit number xyz divisible by 7?

(1) |2*z - 10x - y| is divisible by 7
(2) (z + 3y + 2x) is divisible by 7

That is very interesting problem. I choose B because I cannot figure out how to play with (1).
Thanks for the problem and the solution.
Re: Is a three-digit number xyz divisible by 7?   [#permalink] 27 Aug 2014, 16:23
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