There are N students in a class. When the students are distributed into groups that contain 4A number of students each, 3 students are left without a group. When the students are distributed into groups that containA/3number of students each, no students are left without a group. Which of the following statements is correct?I. If the students are distributed into groups that contain A+ 1 students each, the number of students that are left without a group can be 2II. If the students are distributed into groups that contain 3 students each, no students are left without a groupIII. If the students are distributed into groups that contain 12 students each, 9 students are left without a groupa)I onlyb)II onlyc)III onlyd)I and II onlye)II and III onlyCorrect answer is option 'B'. Can you explain this answer?

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I. If the students are distributed into groups that contain A+ 1 students each, the number of students that are left without a group can be 2

II. If the students are distributed into groups that contain 3 students each, no students are left without a group

III. If the students are distributed into groups that contain 12 students each, 9 students are left without a group

a)
I only
b)
II only
c)
III only
d)
I and II only
e)
II and III only

Correct answer is option 'B'. Can you explain this answer?

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Test: Even And Odd Integers- 1

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There are N students in a class. When the students are distributed int...

Given:

N = (4A)x + 3, where quotient x is an integer
N=(A/3)k

, where k is an integer
Since N denotes the number of students, N must be a positive integer

To find: Which of the 3 statements is/are correct?

Approach:

To answer the question, we first need to get an idea about the values of N and A
Then, we’ll evaluate the 3 statements one by one. A statement will be a must be true statement only if it holds true for all possible values of N and A

Working Out:

Drawing inferences about N and A
- We’re given that N is of the form (4A)x + 3
  - Now, the term (4A)x will be even for all values of A and x
  - So, N is of the form Even + Odd
  - So, N is odd
Also, in the second piece of given information, we’re told that each student group contains A/3 students
- Since number of students cannot be in fractions, this means A must be a multiple of 3

Also, N=(A/3)k
We’ve inferred above that N is odd
So, the product of integers (A/3)
and k is odd
This is possible only if (A/3)
and k themselves are odd
So, (A/3) is odd

Thus, A = (Odd number 3)*(Another odd integer)
So, A is odd as well
Thus, combining our above inferences about A, we can say that A is an odd multiple of 3.
- So, we can write: A = 3(2B+1) = 6B + 3, where B is an integer

Evaluating Statement I
- If the students are distributed into groups that contain A + 1 students each, the number of students that are left without a group can be 2
- A + 1 = (6B+3) + 1 = 6B + 4 = Even number
- Let the number of groups that contain A + 1 students be G.
- (Odd Number N) = (Even Number A+1)*(G) + Remainder
  - So, (Odd Integer) = Even Integer + Remainder
  - So, Remainder = Odd Integer minus Even Integer
- So, the remainder when N is divided by A +1 will be an odd integer.
  - Therefore, the remainder cannot be 2
- So, Statement I is not correct

Evaluating Statement II
- If the students are distributed into groups that contain 3 students each, no students are left without a group
- To answer this question, we need to evaluate if N is divisible by 3 or not
  - If it is, then Statement II is indeed a must be true statement.
- Now, we’ve analyzed above that:
  - N = (4A)x + 3
  - And, A = 6B+3
- So, we can write: N = 4(6B+3)x + 3
  - = 4*3(2B+1)x + 3
  - =3{4(2B+1)x + 1}
- Thus, we see that N is completely divisible by 3
- So, Statement II is correct

Evaluating Statement III
- If the students are distributed into groups that contain 12 students each, 9 students are left without a group
- To evaluate this statement, we need to determine if the remainder when N is divided by 12 is 9
  - If it is, then this is a must be true statement.
- In our analysis of Statement II above, we’ve inferred that:
  - N = 3{4(2B+1)x + 1}
  - Simplifying this: N = 12(2B+1)x + 3
  - So, when N is divided by 12, the first term in the above expression: 12(2B+1)x will be completely divisible by 12
  - So, the remainder when N is divided by 12 will be 3
- Therefore, Statement III is incorrect

Getting to the answer
- From our analysis above, we’ve concluded that only Statement II is always correct.

Looking at the answer choices, we see that the correct answer is Option B

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Most Upvoted Answer

There are N students in a class. When the students are distributed int...

To solve this problem, we need to use algebraic equations to represent the given information and then analyze the resulting equations.

Let's start by assigning variables to the unknowns in the problem:

N: Total number of students in the class
A: Number of students in each group

We are given two conditions:

1. When the students are distributed into groups that contain 4A number of students each, 3 students are left without a group.
This can be represented by the equation N = X * (4A) + 3, where X is the number of groups.

2. When the students are distributed into groups that contain A/3 number of students each, no students are left without a group.
This can be represented by the equation N = Y * (A/3), where Y is the number of groups.

Now, let's analyze these equations and find the values of X and Y:

1. Distributing into groups of 4A students:
N = X * (4A) + 3
N = 4AX + 3

2. Distributing into groups of A/3 students:
N = Y * (A/3)
N = (AY)/3

Now, we can equate these two equations and solve for Y:

4AX + 3 = (AY)/3
12AX + 9 = AY
12X + 9/A = Y

From this equation, we can see that Y is an integer if A is a multiple of 3. This satisfies the condition mentioned in the problem that when distributed into groups of A/3 students, no students are left without a group.

Therefore, the correct statement is:

II. If the students are distributed into groups that contain 3 students each, no students are left without a group.

Hence, the correct answer is option 'B'.

Answered by Talent Skill Learning · View profile

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There are N students in a class. When the students are distributed into groups that contain 4A number of students each, 3 students are left without a group. When the students are distributed into groups that containA/3number of students each, no students are left without a group. Which of the following statements is correct?I. If the students are distributed into groups that contain A+ 1 students each, the number of students that are left without a group can be 2II. If the students are distributed into groups that contain 3 students each, no students are left without a groupIII. If the students are distributed into groups that contain 12 students each, 9 students are left without a groupa)I onlyb)II onlyc)III onlyd)I and II onlye)II and III onlyCorrect answer is option 'B'. Can you explain this answer?

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