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If x, y and z are positive integers such that when x is divi

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If x, y and z are positive integers such that when x is divi [#permalink] New post   09 May 2021, 02:27
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If x, y and z are positive integers such that when x is divided by y, the remainder is z and when x is divided by z, the remainder is (y - 2), then which of the following must be true?
(A) z = (y-1)
(B) (x + z) is divisible by y
(C) (x-1) is divisible by y
(D) y is even
(E) None of these

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E
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Re: If x, y and z are positive integers such that when x is divi [#permalink] New post   09 May 2021, 02:58
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The tough one! My answer is A.

For example: x=5 and y=3, x/y --> Remainder=z=2; x/z=5/2 --> Remainder=1=y-2

So z=2 and y=3, which means z=y-1

This satisfies only option A. I tried different values but can not eliminate the option A.
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Re: If x, y and z are positive integers such that when x is divi [#permalink] New post   19 Jul 2021, 12:45
2 also fills up and it is universal true
when x/y |R=z

X+Z must be divisible by y

easy example is 5/2 r=1
5+1 is divisible by 2 try it
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Re: If x, y and z are positive integers such that when x is divi [#permalink] New post   09 Aug 2021, 00:24
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Tough one.. I don't the answer yet, but looks like option E.

Here is how it goes..

y when divided by x gives remainder z, so we can write equation
kx = y - z, where k=quotient --------- (I)

when x is divided by z, it gives remainder y-2, so we can write
kz = x - (y-2), where k=quotient --------- (II)

if we do (I) + (II), we get:
k(x+z) = x - (z-2)
so, we can say if x is divided by (x+z) we get remainder z-2

if we do (I) - (II), we get:
k(x-z) = 2*(y-1) - (x+z)
so, we can say if 2*(y-1) is divided by (x-z) we get remainder (x+z)
Alternatively, if (x+z) is divided by (z-x) we get remainder 2(y-1)

In any such case, it does not go with the option choices provided
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Re: If x, y and z are positive integers such that when x is divi [#permalink] New post   09 Aug 2021, 09:08
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If x divided by y leaves a remainder of z, then y > z. And if x divided by z leaves a remainder of y - 2, then z > y - 2. Therefore, we have y > z > y - 2, and since x, y, and z are positive integers z = y - 1. Is this reasoning correct?

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Re: If x, y and z are positive integers such that when x is divi [#permalink] New post   10 Aug 2021, 13:22
I got "A" as an answer. Can someone please clarify where my logic is incorrect?

(1) "when x is divided by y, the remainder is z"
from this, we can write that y>z (remainder is always smaller than the divider)


(2) "when x is divided by z, the remainder is (y - 2)"
from this, we can write that Y-2<Z, or we can rewrite is as Y<Z+2

(1)+(2)
Z<Y<Z+2
but x, y and z are positive integers.
so Y must be equal to Z+1.

Any suggestions why it is wrong?
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Re: If x, y and z are positive integers such that when x is divi [#permalink] New post   06 Oct 2022, 13:03
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Re: If x, y and z are positive integers such that when x is divi [#permalink]
06 Oct 2022, 13:03
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