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W, X, Y and Z are four different positive integers.
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Updated on: 12 Mar 2019, 07:54
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16% (02:38) correct
83% (02:45) wrong based on 77 sessions
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W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W. If W = X – 4, which of the following is a possible value of X?
W, X, Y and Z are four different positive integers.
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12 Mar 2019, 07:58
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GreenlightTestPrep wrote:
W, X, Y and Z are four different positive integers. When X is divided by Y, the quotient is Z and the remainder is W. If W = X – 4, which of the following is a possible value of X?
Indicate ALL possible values of X.
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8 (F) 9 (G) 10 (H) 11
Given: When X is divided by Y, the quotient is Z and the remainder is W. ------ASIDE------- There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R" For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2 Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3 --------------------------------------- So, from the given information, we can write: X = YZ + W
Also given: W = X – 4 Take X = YZ + W and replace X with X - 4 to get: X = YZ + X - 4 Subtract X from both sides: 0 = YZ - 4 Rewrite as: 4 = YZ
We're told that Y and Z are DIFFERENT positive INTEGERS. So, there are only 2 possible cases: case i: Y = 1 and Z = 4 case ii: Y = 4 and Z = 1
case i yields a CONTRADICTION. If Y = 1, then we are dividing X by 1, and if we divide by 1, the remainder will always be ZERO. In other words, if Y = 1, then W = 0 However, we are told that W is a POSITIVE integer. So, we can definitely rule out case i, which means it MUST be the case that Y = 4 and Z = 1 (case ii)
So, we have: When X is divided by 4, the quotient is 1 and the remainder is W This tells us that 4 divides into X 1 time So, the possible values of X are: 4, 5, 6, and 7 (since 4 divides into each value 1 time). Let's check each case.
If X = 4, then the remainder (W) is 0. Since we're told W is POSITIVE, we can ELIMINATE this case. If X = 5, then the remainder (W) is 1. Since Y already equals 1, and since we're told the numbers are DIFFERENT, we can ELIMINATE this case. If X = 6, then the remainder (W) is 2. So, when X (6) is divided by 4 (Y), the quotient (Z) is 1, and the remainder (W) is 2. PERFECT! If X = 7, then the remainder (W) is 3. So, when X (7) is divided by 4 (Y), the quotient (Z) is 1, and the remainder (W) is 3. PERFECT!
Re: W, X, Y and Z are four different positive integers.
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29 Jun 2020, 19:16
Why in the second case answer can't be 5? Because previously you outlined that the case Y=1 and X=4 is incorrect since when Y is divided by 1 remainder always 0, and now you say that 5 is inviavle, cause Y is already 1? However we should only consider the second case where Y is predetermined as 4? Could you please explain?
Re: W, X, Y and Z are four different positive integers.
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20 May 2022, 22:33
2
Remember: Remainder W will always be lesser than the divisor Y. It's a general remainder rule, and very powerful when it comes to quickly solving remainder problems.
What are possible values of W if YZ=4?.... Based on above property, W can be 1, 2, 3. But 1 is not possible, so W can be 2 and 3.
Now look for answer choices that give such value of X so that W could be wither 2 or 3. Hence answer choices C and D.
Re: W, X, Y and Z are four different positive integers.
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06 Apr 2025, 17:31
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Re: W, X, Y and Z are four different positive integers. [#permalink]
Updated on: 12 Mar 2019, 07:54
16% (02:38) correct
