If z is an integer such that ||z - 30| - 43| = 62 which of the followi

Question

If z is an integer such that ||z - 30| - 43| = 6^2 which of the following could be value of |r|, where r is the remainder obtained when z is divided by 7?

I. 0
II. 2
III. 4

A I and II only

B I and III only

C II and III only

D I, II and III

E None of the above

IanStewart · Accepted Answer

I don't understand why there are pointless complications in the question. Remainders are never negative, so there's no reason to ask for "|r|", since that is always equal to r. Nor is there any reason to write 6^2 instead of 36 in the original absolute value equation. They'd only do that in a real GMAT question if they were trying to suggest to a test taker that there would be some advantage to thinking about that number as a square, instead of as 36, but there is no reason to think about 36 as if it were "6^2" in this problem.

Since |a - b| is the distance between a and b on the number line, the original absolute value equation tells us that |z-30| and 43 are separated by 36 on the number line. So either |z-30| is 36 greater than 43, and is equal to 79, or |z-30| is 36 less than 43, and is equal to 7. So we have two equations, and again using distances (or whatever absolute value method you prefer) :

|z - 30| = 79 --> so z is either 79 less than or 79 greater than 30, and z = -49 or 109
|z - 30| = 7 --> so z is either 7 less than or 7 greater than 30, and z = 23 or 37

We can now divide each solution by 7 and see what remainders are possible. You will absolutely never need to calculate remainders when dividing negative numbers by anything on the GMAT, so the first solution, z = -49, is one you'd never get in a GMAT question about remainders. But since that number is divisible by 7, it gives a remainder of 0. The number 109 is 105 + 4, and since 105 is divisible by 7, we get a remainder of 4. Since 23 and 37 both give a remainder of 2 when we divide by 7, the answer is I, II and III.

Kinshook · Answer

Given: z is an integer such that ||z - 30| - 43| = 6^2

Asked: Which of the following could be value of |r|, where r is the remainder obtained when z is divided by 7?

|z-30| = 43+-36 => |z-30| = 7 or 79
If |z-30| = 7 => z = 30 +-7 => z = 37 or 23
If |z-30| = 79 => z = 30 +-79 => z = 109 or -49
z= {37,23,109,-49}
Remainder r = {2,2,4,0} = {2,4,0}

IMO D

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem ↧↧↧

https://www.youtube.com/watch?v=G7bg0f4EWSs

Given that z is an integer such that ||z - 30| - 43| =  6^2  and we need to find which of the following could be value of |r|, where r is the remainder obtained when z is divided by 7

Given that r is the remainder
=> r will be non-negative
=> |r| = r

Let's simplify ||z - 30| - 43| =  6^2  = 36

We will have two cases

-  Case 1: |z-30| - 43 = +36

=> |z-30| = 36 + 43 = 79
=> z-30 = +79 or z-30 = -79
=> z = 79 + 30 or z = -79 + 30
=> z = 109 or -49

Remainder(r) when 109 is divided by 7 = 4
Remainder(r) when -49 is divided by 7 = 0

-  Case 2: |z-30| - 43 = -36

=> |z-30| = -36 + 43 = 7
=> z-30 = +7 or z-30 = -7
=> z = 7 + 30 or z = -7 + 30
=> z = 37 or 23

Remainder(r) when 37 is divided by 7 = 2
Remainder(r) when 23 is divided by 7 = 2

So,  Remainder(r) can be 0, 2, 4

So,  Answer will be D 
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

https://www.youtube.com/watch?v=huFiBk8PaGY

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If z is an integer such that ||z - 30| - 43| = 62 which of the followi

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

We will have two cases
-Case 1: \|z-30\| - 43 = +36 => \|z-30\| = 36 + 43 = 79 => z-30 = +79 or z-30 = -79 => z = 79 + 30 or z = -79 + 30 => z = 109 or -49 Remainder(r) when 109 is divided by 7 = 4 Remainder(r) when -49 is divided by 7 = 0	-Case 2: \|z-30\| - 43 = -36 => \|z-30\| = -36 + 43 = 7 => z-30 = +7 or z-30 = -7 => z = 7 + 30 or z = -7 + 30 => z = 37 or 23 Remainder(r) when 37 is divided by 7 = 2 Remainder(r) when 23 is divided by 7 = 2

GMAT Problem Solving (PS) Questions

If z is an integer such that ||z - 30| - 43| = 62 which of the followi

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards