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aragonn
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For positive integers 10 < K < Z < Q, what is the value of the remaind 02 Dec 2018, 01:55
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

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78% (01:54) correct 22% (01:53) wrong based on 383 sessions

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For positive integers 10 < K < Z < Q, what is the value of the remainder when Q is divided by Z?

(1) Q = K^2

(2) Z = K + 1
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C
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For positive integers 10 < K < Z < Q, what is the value of the remaind 02 Dec 2018, 10:06
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From statement 1:

Q = \(K^2\)
From the question stem, Min value of K is 11 as K is an integer.
But no info about Z. Insufficient.

From statement 2:

Z = K+1
Min value of K is 11. Then Z is 12.
but Q can be any integer, hence insufficient.

Combining both,

\(K = Z-1\)
\(Q = K^2\)
\(Q = Z^2+1-2Z\).
When Q is divided by Z, The remainder is always as \(Z^2\) and -2Z is always divisible by Z.

C is the asnwer.
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For positive integers 10 < K < Z < Q, what is the value of the remaind 02 Dec 2018, 16:43
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Afc0892

When Q is divided by Z, The remainder is always as \(Z^2\) and -2Z is always divisible by Z.

C is the asnwer.
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To quickly clarify, I think you have it right except for this line.

When we take Q=\(Z^2\)+1−2ZQ and divide it by Z we get Z-2 R 1. The remainder is due to the +1 term divided by Z as we know the other are multiples of Z and Z is not one.
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For positive integers 10 < K < Z < Q, what is the value of the remaind Updated on: 03 Dec 2018, 05:29
Originally posted by MahmoudFawzy on 03 Dec 2018, 04:56.
Last edited by MahmoudFawzy on 03 Dec 2018, 05:29, edited 1 time in total.
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aragonn

Official Explanation:



For this one, we’ll at least start out with picking numbers and see how far we get with this strategy.

Statement #1: Q = \(K^2\)

For simplicity, say that K = 20, so Q = 400. If Z = 40, then it goes evenly into 400, so the remainder is zero. On the other hand, if Z = 250, then it goes once into 400 with a remainder of 150. Two different choices of numbers give two different values of the remainder. This choice, alone and by itself, is not sufficient.

Statement #2: Z = K + 1

For simplicity, let K = 10. This makes Z = 11, but that doesn’t matter. Under this statement, there’s no restriction on the value of Q. If Q = 60, then 10 divides evenly into 60, and the remainder is zero. If Q = 69, then 10 goes into it six times with a remainder of 9. Two different choices of numbers give two different values of the remainder. This choice, alone and by itself, is not sufficient.

Combined statements:

Let’s pick K = 11. Then Z = 12 and Q = 121. We know that 12 goes evenly into 120, so when we divide 121 by 12, we get a remainder of 1.

It gets hard to pick numbers and do the squaring & division without a calculator. All other choices will result in the same remainder of 1. Remember that we can use picking numbers to disqualify an answer, to show that it leads to two different conclusions, but if the same numerical answer results every time, we need to verify that with algebra or logic.

Remember the Difference of Two Squares pattern. We know that

Q − 1 = \(K^2\)− 1 = (K + 1)(K − 1)

Thus, Z = K + 1 always divides evenly into Q − 1. If we add one, the next integer up from Q − 1 is Q—the integer Q is exactly one more than the integer Q − 1. When we divide Q by Z = K + 1, Z always goes evenly into Q − 1, and the extra 1 is left over, so we always will have a remainder of 1.

The two statements together allow us to find a unique numerical answer to the prompt questions. Combined, the statements are sufficient.

Answer = (C)
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Thanks So much.
I got it totally wrong.

It is C
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For positive integers 10 < K < Z < Q, what is the value of the remaind 03 Dec 2018, 05:22
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Official Explanation:



For this one, we’ll at least start out with picking numbers and see how far we get with this strategy.

Statement #1: Q = \(K^2\)

For simplicity, say that K = 20, so Q = 400. If Z = 40, then it goes evenly into 400, so the remainder is zero. On the other hand, if Z = 250, then it goes once into 400 with a remainder of 150. Two different choices of numbers give two different values of the remainder. This choice, alone and by itself, is not sufficient.

Statement #2: Z = K + 1

For simplicity, let K = 10. This makes Z = 11, but that doesn’t matter. Under this statement, there’s no restriction on the value of Q. If Q = 60, then 10 divides evenly into 60, and the remainder is zero. If Q = 69, then 10 goes into it six times with a remainder of 9. Two different choices of numbers give two different values of the remainder. This choice, alone and by itself, is not sufficient.

Combined statements:

Let’s pick K = 11. Then Z = 12 and Q = 121. We know that 12 goes evenly into 120, so when we divide 121 by 12, we get a remainder of 1.

It gets hard to pick numbers and do the squaring & division without a calculator. All other choices will result in the same remainder of 1. Remember that we can use picking numbers to disqualify an answer, to show that it leads to two different conclusions, but if the same numerical answer results every time, we need to verify that with algebra or logic.

Remember the Difference of Two Squares pattern. We know that

Q − 1 = \(K^2\)− 1 = (K + 1)(K − 1)

Thus, Z = K + 1 always divides evenly into Q − 1. If we add one, the next integer up from Q − 1 is Q—the integer Q is exactly one more than the integer Q − 1. When we divide Q by Z = K + 1, Z always goes evenly into Q − 1, and the extra 1 is left over, so we always will have a remainder of 1.

The two statements together allow us to find a unique numerical answer to the prompt questions. Combined, the statements are sufficient.

Answer = (C)
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For positive integers 10 < K < Z < Q, what is the value of the remaind 03 Dec 2018, 16:03
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aragonn
For positive integers 10 < K < Z < Q, what is the value of the remainder when Q is divided by Z?

Statement #1: Q = \(K^2\)

Statement #2: Z = K + 1
Show more



Neither statement alone is going to help too much! Statement 1 doesn't tell us the value of Z. Statement 2 doesn't tell us the value of Q. You can't be 100% confident that each statement is insufficient just on that basis, but it's a good clue. Here's how I convinced myself:

Statement 1: Suppose that Z is only 1 smaller than Q. In that case, the remainder of Q/Z should be 1. But if Z is 2 smaller than Q, the remainder should be 2. It's definitely possible for Z to be either of those numbers, since Q = K^2, which is much bigger than K - so there's a lot of 'wiggle room' for Z to have different values in between K and K^2.

Statement 2: If Q is only 1 bigger than Z, the remainder is 1. If Q is 2 bigger than Z, the remainder is 2. Testing cases with numbers that are close together is useful for remainders! It's much easier to calculate the remainder of 99999999/99999998 than the remainder of 99999999/375. Not sufficient.

Putting the two statements together, I'd like to test cases, since the math is getting a bit intense. Let's start with the easiest number we're allowed to use: K = 11.
In that case, Z = 12 and Q = 11^2, which is 121. 121/12 has a remainder of 1.

If K = 12, Z = 13 and Q = 12^2 = 144. 144/13 has a remainder of 1 as well.

At this point, you could either choose C and move on, or you could try to come up with a mathematical justification to choose C. I thought about it like this: K goes into Q exactly K times, with a remainder of 0. Since Z is slightly bigger than K, but only by a tiny bit, we should be able to fit it into Q 1 fewer time. So, K+1 should go into Q K-1 times. That gives you (K+1)(K-1) = K^2-1, which has 1 left over when dividing into K^2.
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For positive integers 10 < K < Z < Q, what is the value of the remaind 23 Nov 2020, 19:57
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I don't understand how to solve this one... how do you just seemingly pick the perfect #s to try at random?! I never would've thought to substitute 11 for k, 12 for z, and 121 for Q. Those would never have just jumped into my mind.

Even doing it algebraically, apparently you -1 +1 from the numerator... how in the world would one arrive at that strategy to solve this problem?!
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For positive integers 10 < K < Z < Q, what is the value of the remaind 08 Aug 2022, 03:10
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I solved this using the co-prime concept

From statement 1:

\(Q = K^2\)
From the question stem, Min value of K is 11 as K is an integer. No info about Z. Insufficient.

From statement 2:

\(Z = K+1\)
Min value of K is 11 also K and Z are adjacent integers. In other words, K and Z are co-prime and have no common factors between them but Q can be any integer, hence Insufficient.

Combining both,

Lets say \(K\) has two prime factors \(a*b\). So Z will have prime factors other than \(a * b\) say \(c * d\)

As \(Q=K^2\) we get \(Q=a^2*b^2\)

So \(Q = K^2\) can never be divisible by \(Z\) as \( K^2\) introduces no new factors and \(K\) has no common factors with \(Z\)

C is the answer.
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For positive integers 10 < K < Z < Q, what is the value of the remaind 24 Dec 2022, 02:17
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For positive integers 10 < K < Z < Q, what is the value of the remainder when Q is divided by Z?

(1) Q = K^2

(2) Z = K + 1

I used Plugging in Numbers (PIN) picked multiples of 10 for values of K & Q

Fact 1: q = k^2 Not Sufficient
Fact 2 : Z = K + 1 Not Sufficient

Combining 10<K<K + 1<K^2
Say 10<20<21<20^2 = 400/21 - Remainder = 1 Tested 20 for even
Say 10<15<16<15^2 = 225/16 - Remainder =1 Tested 15 for odd

Hence Sufficient
C
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For positive integers 10 < K < Z < Q, what is the value of the remaind 14 Apr 2024, 23:18
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