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DIRECTIONS for the question: Solve the following question and mark the best possible option.
A positive integer N can be expressed as the sum of 9 consecutive integers, also as the sum of 10 consecutive integers and also as the sum of 11 consecutive integers. What is the smallest value of N?
  • a)
    518
  • b)
    495
  • c)
    760
  • d)
    990
Correct answer is option 'B'. Can you explain this answer?
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DIRECTIONSfor the question:Solve the following question and mark the b...
Suppose N = S9 = n + (n + 1) + (n + 2)….. + (n + 8) = 9(n + 4).
N can also be written as :
S10 = a + (a + 1) + … + (a + 9) = 5(2a + 9) or S11 = x + (x + 1) + … + (x + 10) = 11(x + 5).
Since N = S9 = S10 = S11, we know that N must be divisible by 9, 5 and 11.
So, the smallest value of N must be LCM (5, 9, 11) = 495.
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DIRECTIONSfor the question:Solve the following question and mark the b...
Question Analysis:
We are given that a positive integer N can be expressed as the sum of 9 consecutive integers, 10 consecutive integers, and 11 consecutive integers. We need to find the smallest value of N.

Solution:
Let's assume that the middle term of the 9 consecutive integers is x.
So, the 9 consecutive integers will be (x-4), (x-3), (x-2), (x-1), x, (x+1), (x+2), (x+3), and (x+4).

Similarly, let's assume that the middle term of the 10 consecutive integers is y.
So, the 10 consecutive integers will be (y-4), (y-3), (y-2), (y-1), y, (y+1), (y+2), (y+3), (y+4), and (y+5).

Also, let's assume that the middle term of the 11 consecutive integers is z.
So, the 11 consecutive integers will be (z-5), (z-4), (z-3), (z-2), (z-1), z, (z+1), (z+2), (z+3), (z+4), and (z+5).

Sum of 9 consecutive integers:
The sum of 9 consecutive integers can be calculated using the formula:
Sum = Number of terms * (First term + Last term) / 2

In this case, the sum of 9 consecutive integers will be:
N = 9 * (x-4 + x+4) / 2
= 9 * (2x) / 2
= 9x

Sum of 10 consecutive integers:
The sum of 10 consecutive integers can be calculated using the formula:
Sum = Number of terms * (First term + Last term) / 2

In this case, the sum of 10 consecutive integers will be:
N = 10 * (y-4 + y+5) / 2
= 10 * (2y + 1) / 2
= 5(2y + 1)

Sum of 11 consecutive integers:
The sum of 11 consecutive integers can be calculated using the formula:
Sum = Number of terms * (First term + Last term) / 2

In this case, the sum of 11 consecutive integers will be:
N = 11 * (z-5 + z+5) / 2
= 11 * (2z) / 2
= 11z

Comparing the sums:
From the above calculations, we can see that N = 9x = 5(2y + 1) = 11z.

Let's find the smallest value of N by comparing the sums of the three consecutive integers.

Comparing N = 9x and N = 5(2y + 1):
Since N is a positive integer, both 9x and 5(2y + 1) should be positive integers.
The smallest positive integer value for 9x is 9, and for 5(2y + 1) is 5
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Directions for Questions: The first-year students in a business school are split into six sections. In 2019 the Business Statistics course was taught in these six sections by Annie, Beti, Chetan, Dave, Esha, and Fakir. All six sections had a common midterm (MT) and a common end term (ET) worth 100 marks each. ET contained more questions than MT. Questions for MT and ET were prepared collectively by the six faculty members. Considering MT and ET together, each faculty member prepared the same number of questions.Each of MT and ET had at least four questions that were worth 5 marks, at least three questions that were worth 10 marks, and at least two questions that were worth 15 marks. In both MT and ET, all the 5-mark questions preceded the 10-mark questions, and all the 15- mark questions followed the 10-mark questions.The following additional facts are known.i. Annie prepared the fifth question for both MT and ET. For MT, this question carried 5 marks.ii. Annie prepared one question for MT. Every other faculty member prepared more than one questions for MT.iii. All questions prepared by a faculty member appeared consecutively in MT as well as ET.iv. Chelan prepared the third question in both MT and ET: and Esha prepared the eighth question in both.v. Fakir prepared the first question of MT and the last one in ET. Dave prepared the last question of MT and the first one in ET.(2019)Q. Which of the following questions did Beti prepare in ET?

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The first year students in a business school are split into six sections. In 2019 the Business Statistics course was taught in these six sections by Annie, Beti, Chetan, Dave, Esha, and Fakir. All six sections had a common midterm (MT) and a common end term (ET) worth 100 marks each. ET contained more questions than MT. Questions for MT and ET were prepared collectively by the six faculty members.Considering MT and ET together, each faculty member prepared the same number of questions.Each of MT and ET had at least four questions that were worth 5 marks, at least three questions that were worth 10 marks, and at least two questions that were worth 15 marks. In both MT and ET, all the 5-mark questions preceded the 10-mark questions, and all the 15-mark questions followed the 10-mark questions.The following additional facts are known. i. Annie prepared the fifth question for both MT and ET. For MT, this question carried 5 marks. ii. Annie prepared one question for MT. Every other faculty member prepared more than one questions for MT.iii. All questions prepared by a faculty member appeared consecutively in MT as well as ET. iv. Chetan prepared the third question in both MT and ET; and Esha prepared the eighth question in both. v. Fakir prepared the first question of MT and the last one in ET. Dave prepared the last question of MT and the first one in ET.Q. Which of the following questions did Beti prepare in ET?

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