The students in a school were to be distributed into rows for an assembly. The headmaster earlier thought of distributing the students such that each row had 10 students, but decided against it since this distribution would have led to the last row having only 5 students. Instead, he distributed the students into rows of equal size, with each row having 21 students. Which of the following could be the number of rows in the assembly?I. 15II. 30III. 21a)I onlyb)II onlyc)III onlyd)I and II onlye)II and III onlyCorrect answer is option 'A'. Can you explain this answer?

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The students in a school were to be distributed into rows for an assembly. The headmaster earlier thought of distributing the students such that each row had 10 students, but decided against it since this distribution would have led to the last row having only 5 students. Instead, he distributed the students into rows of equal size, with each row having 21 students. Which of the following could be the number of rows in the assembly?
I. 15
II. 30
III. 21

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

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

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The students in a school were to be distributed into rows for an assem...

Given:

Let the total number of students be n
When n is divided by 10, the remainder is 5
- So, n is of the form 10k + 5, where k is a non-negative integer
  - k is a non-negative integer because for negative values of k, n will be negative as well, which is not allowed since the number of students cannot be negative.
When n is divided by 21, the remainder is 0
- So, n is also of the form 21j, where j is a positive integer

To Find: Can number of rows be 15, 21 or 30?

Approach:

Number of rows =n/21
1. So, to answer the question about the possible values for number of rows, we need to know the possible values of n.
One way to answer this question is to calculate the value of n corresponding to each of the 3 values for number of rows (15, 21 or 30) and then check these values of n for the constraints on value of n (n is of the form 10k + 5 and 21j)
Another way to answer is to first deduce a general expression for n that satisfies both the given constraints on n. Then, deduce a corresponding general expression for ‘number of rows’. This way, we’ll be able to see which of the 3 values satisfies the general expression for ‘number of rows’. We will illustrate this second approach in the working out section.

Working out:

Finding a general expression for n
- The 2 expressions given about n are:
  - n = 10k + 5 = 5(2k + 1)
    - This means, n is the product of 5 and a positive odd integer
  - And, n = 21j = 3*7j
    - This means, 3 and 7 are definitely the prime factors of n
- So, combining both expressions, we can write:
  - n = 3*5*7h, where h is a positive odd integer
- So, from the general expression of n, we learn that n is odd and definitely divisible by 3, 5 and 7
Finding a general expression for ‘number of rows’
- Number of rows = where h is a positive odd integer
- Thus, the ‘number of rows’ is an odd multiple of 5

Evaluating the 3 values
- Looking at the 3 values, we see that only value I satisfies the inference that ‘number of rows’ is an odd multiple of 5

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

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