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When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?
- a)2
- b)4
- c)8
- d)10
- e)20
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
When a class of n students is divided into groups of 6 students each, ...
Given:
- n = 6k + 2 = 8p + 4, where k and p are non-negative integers (if k or p were negative, then n would be negative too and that’s not possible since the number of students cannot be negative)
To find: The smallest number that can be added to or subtracted from n to make the resulting number divisible by 12
Approach:
1. Let the number to be added or subtracted from n be x. This means,
- Either (students are added) n + x = 12j, where j is a positive integer
- This means, n = 12j - x
- Or (students are removed) n – x = 12m, where m is a positive integer
- This means, n = 12m + x
2. The expressions n = 12j – x and n = 12m + x make us realize that in order to answer this question, we need to know the remainder when n is divided by 12
- Say the remainder is 3 (That is, n = 12m + 3). In this case, the smallest change to make the number of students divisible by 12 is to remove 3 students from the class.
- Same will be the case for any value of remainder less than or equal to 6
- Say the remainder is 10 (That is, n = 12m + 10). In this case, the smallest change to make the number of students divisible by 12 is to add 2 students to the class
- Same will be the case for any value of remainder greater than 6
3. Let the remainder that n leaves when divided by 12 be r. So, our Goal expression is: n = 12q + r, where quotient q is an integer and 0 ≤ r < 12
4. We’ll use the given information about n to drive towards our Goal expression
Working Out:
- n = 6k + 2 = 8p + 4
- LCM (6, 8) = 24
- This means, n is a number of the form 24z + r’, where 0 ≤ r’ < 24
- When 24z + r’ is divided by 6, the remainder is 2
- In this expression, the term 24z is divisible by 6
- So, possible values of r’ that can lead to remainder 2 = {2, 8, 14, 20}
- When 24z + r’ is divided by 6, the remainder is 2
- This means, n is a number of the form 24z + r’, where 0 ≤ r’ < 24
- LCM (6, 8) = 24
- When 24z + r’ is divided by 8, the remainder is 4
- In this expression, the term 24z is divisible by 8
- So, possible values of r’ that can lead to remainder 4 = {4, 12, 20}
- The only value of r’ that satisfies both these conditions = {20}
- So, r’ = 20
- Therefore, n = 24z +20
- Let’s now rearrange the above equation a little bit to make it comparable to our GOAL expression: n = {24z + 12} + 8
- Now this form is exactly comparable to our Goal Expression
- By comparison, we see that Remainder = 8
- Since remainder is greater than 6, the smallest change to make the number of students divisible by 12 is to add 4 students.
Looking at the answer choices, we see that the correct answer is Option B
Most Upvoted Answer
When a class of n students is divided into groups of 6 students each, ...
Problem Analysis:
Let's assume the total number of students in the class is 'x'.
Given that when the class is divided into groups of 6 students each, 2 students are left without a group. This implies that 'x' is 2 more than a multiple of 6.
So, we can write the equation as:
x = 6a + 2, where 'a' is a positive integer.
Similarly, when the class is divided into groups of 8 students each, 4 students are left without a group. This implies that 'x' is 4 more than a multiple of 8.
So, we can write the equation as:
x = 8b + 4, where 'b' is a positive integer.
Calculating the values of 'a' and 'b':
We can solve the above equations to find the values of 'a' and 'b'.
Equation 1: x = 6a + 2
Equation 2: x = 8b + 4
To find the smallest value of 'x' that satisfies both equations, we can equate the right-hand sides of the equations and solve for 'a' and 'b'.
6a + 2 = 8b + 4
Rearranging the equation:
6a - 8b = 2
Now, let's check for the smallest positive values of 'a' and 'b' that satisfy this equation.
When a = 2, b = 1, the equation becomes:
6(2) - 8(1) = 12 - 8 = 4
When a = 4, b = 2, the equation becomes:
6(4) - 8(2) = 24 - 16 = 8
We can observe that as 'a' and 'b' increase, the value of 'x' also increases. So, the smallest value of 'x' that satisfies both equations is 4.
Calculating the number of students to be added or removed:
To find the number of students to be added or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each, we need to calculate the remainder when 'x' is divided by 12.
x = 6a + 2
For a = 1, x = 6 + 2 = 8 (remainder = 8 % 12 = 8)
For a = 2, x = 12 + 2 = 14 (remainder = 14 % 12 = 2)
For a = 3, x = 18 + 2 = 20 (remainder = 20 % 12 = 8)
For a = 4, x = 24 + 2 = 26 (remainder = 26 % 12 = 2)
From the above calculations, we can see that 'x' has a remainder of 8 or 2 when divided by 12.
To make 'x' divisible by 12, we need to add or remove the minimum number of students that will result in a remainder of 0 when 'x' is divided by 12.
Since we already have 4 students in the class, we need to add 8 students to make the total number of students divisible by
Let's assume the total number of students in the class is 'x'.
Given that when the class is divided into groups of 6 students each, 2 students are left without a group. This implies that 'x' is 2 more than a multiple of 6.
So, we can write the equation as:
x = 6a + 2, where 'a' is a positive integer.
Similarly, when the class is divided into groups of 8 students each, 4 students are left without a group. This implies that 'x' is 4 more than a multiple of 8.
So, we can write the equation as:
x = 8b + 4, where 'b' is a positive integer.
Calculating the values of 'a' and 'b':
We can solve the above equations to find the values of 'a' and 'b'.
Equation 1: x = 6a + 2
Equation 2: x = 8b + 4
To find the smallest value of 'x' that satisfies both equations, we can equate the right-hand sides of the equations and solve for 'a' and 'b'.
6a + 2 = 8b + 4
Rearranging the equation:
6a - 8b = 2
Now, let's check for the smallest positive values of 'a' and 'b' that satisfy this equation.
When a = 2, b = 1, the equation becomes:
6(2) - 8(1) = 12 - 8 = 4
When a = 4, b = 2, the equation becomes:
6(4) - 8(2) = 24 - 16 = 8
We can observe that as 'a' and 'b' increase, the value of 'x' also increases. So, the smallest value of 'x' that satisfies both equations is 4.
Calculating the number of students to be added or removed:
To find the number of students to be added or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each, we need to calculate the remainder when 'x' is divided by 12.
x = 6a + 2
For a = 1, x = 6 + 2 = 8 (remainder = 8 % 12 = 8)
For a = 2, x = 12 + 2 = 14 (remainder = 14 % 12 = 2)
For a = 3, x = 18 + 2 = 20 (remainder = 20 % 12 = 8)
For a = 4, x = 24 + 2 = 26 (remainder = 26 % 12 = 2)
From the above calculations, we can see that 'x' has a remainder of 8 or 2 when divided by 12.
To make 'x' divisible by 12, we need to add or remove the minimum number of students that will result in a remainder of 0 when 'x' is divided by 12.
Since we already have 4 students in the class, we need to add 8 students to make the total number of students divisible by
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When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?a)2b)4c)8d)10e)20Correct answer is option 'B'. Can you explain this answer?
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When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?a)2b)4c)8d)10e)20Correct answer is option 'B'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?a)2b)4c)8d)10e)20Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?a)2b)4c)8d)10e)20Correct answer is option 'B'. Can you explain this answer?.
When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?a)2b)4c)8d)10e)20Correct answer is option 'B'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?a)2b)4c)8d)10e)20Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?a)2b)4c)8d)10e)20Correct answer is option 'B'. Can you explain this answer?.
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