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# 100x+10y+z)/5=10w+z if x, y, z,w are all positive integers

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VP
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100x+10y+z)/5=10w+z if x, y, z,w are all positive integers [#permalink]

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25 Jan 2008, 10:55
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(100x+10y+z)/5=10w+z

if x, y, z,w are all positive integers less than 10 that satisfy the above equation, what is the value of w?

1. x=y-6
2. x+y+z=13

Last edited by marcodonzelli on 25 Jan 2008, 11:27, edited 1 time in total.
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25 Jan 2008, 11:21
1
KUDOS
marcodonzelli wrote:
if x, y, z,w are all positive integers less than 10 that satisfy the above equation, what is the value of w?

1. x=y-6
2. x+y+z=13

E.

1) NOT SUF because z and w can be any positive integer <10
2) NOT SUF. We can have the following solutions:

X Y Z
2 3 8
3 4 6
5 6 2

1 and 2 together:

From x=y-6, the only possible solution is that:
y=8, x=2 and z=3
But w can be any other positive number<10, such as 4,5....

So E.
VP
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25 Jan 2008, 11:28
netcaesar wrote:
marcodonzelli wrote:
if x, y, z,w are all positive integers less than 10 that satisfy the above equation, what is the value of w?

1. x=y-6
2. x+y+z=13

E.

1) NOT SUF because z and w can be any positive integer <10
2) NOT SUF. We can have the following solutions:

X Y Z
2 3 8
3 4 6
5 6 2

1 and 2 together:

From x=y-6, the only possible solution is that:
y=8, x=2 and z=3
But w can be any other positive number<10, such as 4,5....

So E.

I am sorry, I missed to write down the equation. look at the Q now
CEO
Joined: 17 Nov 2007
Posts: 3580
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Schools: Chicago (Booth) - Class of 2011
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25 Jan 2008, 13:48
1
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Expert's post
A

$$10w+z=\frac{100x+10y+z}{5}$$

100x and 10y are divisible by 5. Therefore, z is also divisible 5 and equal z=5

$$10w+5=\frac{100x+10y+5}{5}$$

$$w=2x+\frac{1}{5}y-\frac{2}{5}$$

y have to be y=2 or y=7.

1. x=y-6. Only y=7 satisfies this condition. sufficient.

2. x+y+5=13 ==> x+y=8. both y=2 and y=7 satisfy condition. insufficient.
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VP
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08 Feb 2008, 08:30
walker wrote:
A

$$10w+z=\frac{100x+10y+z}{5}$$

100x and 10y are divisible by 5. Therefore, z is also divisible 5 and equal z=5

$$10w+5=\frac{100x+10y+5}{5}$$

$$w=2x+\frac{1}{5}y-\frac{2}{5}$$

y have to be y=2 or y=7.

1. x=y-6. Only y=7 satisfies this condition. sufficient.

2. x+y+5=13 ==> x+y=8. both y=2 and y=7 satisfy condition. insufficient.

are you sure y can be 2?
CEO
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Posts: 3580
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08 Feb 2008, 09:21
Expert's post
marcodonzelli wrote:
are you sure y can be 2?

$$w=2x+\frac{1}{5}y-\frac{2}{5}$$

$$\frac{1}{5}y-\frac{2}{5}$$ - have to be an integer. It is possible if $$\frac{y}{5}=\frac{2}{5}$$ or $$\frac{y}{5}=\frac{7}{5}$$

What is OA? Am I wrong?
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08 Feb 2008, 11:27
marcodonzelli wrote:
(100x+10y+z)/5=10w+z

if x, y, z,w are all positive integers less than 10 that satisfy the above equation, what is the value of w?

1. x=y-6
2. x+y+z=13

since z > 0, z < 10, and z is divisible by 5 -> z = 5

20 x + 2y + 1 = 10 w + 5

2y + 1 = 5 or 15 -> y = 2 or 7

1) x = y -6 -> y = 7, x = 1, z = 5 and w = 3 -> sufficient
2) x + y + z = 13 -> z = 5, x+y=8
y =2 -> x = 6, w = 12
y = 7 -> x = 1 -> w = 3 -> insufficient

VP
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17 Feb 2008, 23:25
walker wrote:

$$w=2x+\frac{1}{5}y-\frac{2}{5}$$

$$\frac{1}{5}y-\frac{2}{5}$$ - have to be an integer. It is possible if $$\frac{y}{5}=\frac{2}{5}$$ or $$\frac{y}{5}=\frac{7}{5}$$

What is OA? Am I wrong?

note that z must be 5 so that 100x+10y+z is a multiple of 5. since 100x+10y+z=5(10w+5), a multiple of 25, y must be either 2 or 7, as multiples of 25 that do not end in 0 must end in 25 or 75.

1.since alla variables are positive, y must be 7 and x must be 1. suff
2.x+y=8 and as x must be less than 5 for the quotient to have only 2 digits, y must be 7 and x 1. suff
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18 Feb 2008, 12:39
(100X + 10Y + Z ) / 5 = 10W + Z

Z can be 5 or 0 since all variables are integers and 100X + 10Y has to be a multiple of 10.
Since all variables have to be positive, Z = 5

multiplying both sides by 5 to get rid of fraction we have:

100X + 10Y + Z = 50W + 5Z

Since Z = 5,

100X + 10Y + 5 = 50W + 25

100X + 10Y = 50W + 20

Divide by 10 to reduce

10X + Y = 5W + 2

Get W to one side

(10X + Y - 2) / 5 = W

So Since (10X + Y - 2) is divisible by 5 it follows that Y has to be = 7. This is because 10X will always be divisible by 5 and now you need Y-2 to be divisible by 5, the only integers less than 10 that satisfies that is 7 or 2

Since statement 1 is X = Y - 6 , Y must be greater than 6 because all variables are positive (X has to be positive) , so 7 = Y, then by plugging in X = 1 and since you know Z=5 from before we can find W. W=3

Statement 2, after we find out that Z=5 states

X+Y=8

From the stem we know Y=7 or Y=2

If Y=2 X=6

If we plug that into the stem:

(10X + Y - 2) / 5 = W

60/5 = W

W=12 Since W has to be less than 10, we can conclude that Y=7 and X=1.

Choice: D

Definitely wouldnt be able to do this in 2min though
Re: integer equation   [#permalink] 18 Feb 2008, 12:39
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