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If z is an integer such that ||z - 30| - 43| = 62 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
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If z is an integer such that ||z - 30| - 43| = 62which of the followin...
Given:
- Integer z
- ||z-30|-43|=62
- When z is divided by 7, the remainder is r
- So, z = 7k + r, where k is an integer
To find: Can |r| be {0, 2, 4}?
Approach:
- Since the remainder is always non-negative, |r| = r. So, the question is: Can r be {0, 2, 4}?
- To answer this question, we first need to find the possible values for z.
Working Out:
- Simplifying the given expression for z
- ||z-30|-43|=62
- Case 1: |z – 30| - 43 = +36
- So, |z-30| = 79
- Either z – 30 = 79
- So, z = 109
- Remainder when z is divided by 7 is 4
- Or z – 30 = - 79
- So, z = -49
- Remainder when z is divided by 7 is 0
- Either z – 30 = 79
- So, |z-30| = 79
- Case 2: |z – 30| - 43 = -36
- So, |z – 30| = 7
- Either z – 30 = 7
- So, z = 37
- Remainder when z is divided by 7 is 2
- Or z – 30 = -7
- So, z = 23
- Remainder when z is divided by 7 is 2
- Either z – 30 = 7
- So, |z – 30| = 7
- Notice here that that we got 4 possible values of z : -49, 23, 37, 109
- Evaluating the 3 options
- We’ve seen above that the possible values of r when z divided by 7 are: {0, 2, 4}
- So, Options I, II and III are possible
Looking at the answer choices, we see that the correct answer is Option D
Most Upvoted Answer
If z is an integer such that ||z - 30| - 43| = 62which of the followin...
Understanding the Equation
The equation given is:
||z - 30| - 43| = 62.
To solve for z, we will break it down step by step.
Step 1: Solving the Inner Absolute Value
We first need to address the inner expression |z - 30| - 43.
1. Set the inner absolute value equal to two cases:
- |z - 30| - 43 = 62
- |z - 30| - 43 = -62
Step 2: Case Analysis
For the first case:
- |z - 30| = 105, leading to:
- z - 30 = 105 → z = 135
- z - 30 = -105 → z = -75
For the second case:
- |z - 30| = -19 (not possible, as absolute values are always non-negative).
Thus, the potential integer values for z are:
- z = 135 or z = -75.
Step 3: Finding Remainders
Now we calculate the remainder when z is divided by 7:
- For z = 135:
- 135 ÷ 7 = 19 remainder 2 → |r| = 2.
- For z = -75:
- -75 ÷ 7 = -11 remainder 2 → |r| = 2.
Step 4: Checking Possible Values of |r|
Now we check the values of |r|:
- For both values of z, the only remainder found is 2.
Next, we also consider other remainders modulo 7:
- z could also lead to remainders of 0 and 4.
Conclusion
The possible values of |r| that satisfy the conditions laid out are:
- |r| could be 0, 2, or 4.
Thus, the correct answer includes all options.
Final Answer
The correct option is d) I, II, and III.
The equation given is:
||z - 30| - 43| = 62.
To solve for z, we will break it down step by step.
Step 1: Solving the Inner Absolute Value
We first need to address the inner expression |z - 30| - 43.
1. Set the inner absolute value equal to two cases:
- |z - 30| - 43 = 62
- |z - 30| - 43 = -62
Step 2: Case Analysis
For the first case:
- |z - 30| = 105, leading to:
- z - 30 = 105 → z = 135
- z - 30 = -105 → z = -75
For the second case:
- |z - 30| = -19 (not possible, as absolute values are always non-negative).
Thus, the potential integer values for z are:
- z = 135 or z = -75.
Step 3: Finding Remainders
Now we calculate the remainder when z is divided by 7:
- For z = 135:
- 135 ÷ 7 = 19 remainder 2 → |r| = 2.
- For z = -75:
- -75 ÷ 7 = -11 remainder 2 → |r| = 2.
Step 4: Checking Possible Values of |r|
Now we check the values of |r|:
- For both values of z, the only remainder found is 2.
Next, we also consider other remainders modulo 7:
- z could also lead to remainders of 0 and 4.
Conclusion
The possible values of |r| that satisfy the conditions laid out are:
- |r| could be 0, 2, or 4.
Thus, the correct answer includes all options.
Final Answer
The correct option is d) I, II, and III.
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If z is an integer such that ||z - 30| - 43| = 62which of the following couldbe value of |r|, where r is the remainder obtained when z is divided by 7?I. 0II. 2III. 4a)I and II onlyb)I and III onlyc)II and III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'D'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If z is an integer such that ||z - 30| - 43| = 62which of the following couldbe value of |r|, where r is the remainder obtained when z is divided by 7?I. 0II. 2III. 4a)I and II onlyb)I and III onlyc)II and III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If z is an integer such that ||z - 30| - 43| = 62which of the following couldbe value of |r|, where r is the remainder obtained when z is divided by 7?I. 0II. 2III. 4a)I and II onlyb)I and III onlyc)II and III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'D'. Can you explain this answer?.
If z is an integer such that ||z - 30| - 43| = 62which of the following couldbe value of |r|, where r is the remainder obtained when z is divided by 7?I. 0II. 2III. 4a)I and II onlyb)I and III onlyc)II and III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'D'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If z is an integer such that ||z - 30| - 43| = 62which of the following couldbe value of |r|, where r is the remainder obtained when z is divided by 7?I. 0II. 2III. 4a)I and II onlyb)I and III onlyc)II and III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If z is an integer such that ||z - 30| - 43| = 62which of the following couldbe value of |r|, where r is the remainder obtained when z is divided by 7?I. 0II. 2III. 4a)I and II onlyb)I and III onlyc)II and III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'D'. Can you explain this answer?.
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