Question

On the playground, if children are made to stand for drill either 20 to a row or 25 to a row, all rows are complete and no child is left out. What Is the lowest possible number of children in that school?

Answer 1

Hint: Assume the variable of rows if students are arranged 20 per row. Assume other variables of rows if students are arranged 25 per row. Now use the relation between total number of students in both cases to get to conclusion and total number. This total number is the required solution.

Complete step-by-step answer:
Given the condition in the question, can be written as, students are arranged 20 per row or 25 per row, no students left. Let us assume it takes “a” rows if the students are arranged such as 20 students per row to complete all of them. By above statement, we can say total number of students. Total number of students $=20\times a$
Let us assume the total number of students as the variable k. We can write above equation in terms of the variable k as
$k=20\times a$
Let us assume it takes b rows if the students are arranged such as 25 students per row to complete all. We can write an equation in terms of k, given in form of, k = 25b.
So, k is a multiple of 20 and 25, we need the least value of k which is multiple of both 20 and 25. So, we can say the required value k is nothing but the least common multiple of 20, 25. We need prime factorization of both.
Prime factorization: In number theory, prime factorization is the decomposition of a composite number into a product of few prime numbers which are smaller than the original number. This process is carried out by dividing with prime number and finding quotient thus writing number as prime x quotient. Repeat the process for the quotient till you get 1 as the quotient.
By dividing 20 with 2, we can write it in the form of
$20=2\times 10$
By dividing 10 with 2, we can write it in the form of
$20=2\times 2\times 5$
By dividing 5 with 5, we can write it in the form of
$20=2\times 2\times 5\times 1$
We have to stop here as we got 1 as the quotient.
By dividing 25 with 5, we can write it in the form of
$25=5\times 5$
By dividing 5 with 5, we can write it in the form of
$25=5\times 5\times 1$
We have to stop here as we got 1 as the quotient.
Least common multiple: In arithmetic, number theory the least common multiple, lowest common multiple or smallest common multiple is the smallest integer k which is divisible by all the given numbers.
Process to find least common multiple: write all the prime factored forms together. Now find the prime number which is repeating at least one in both or all the numbers. Just combine all the repeating primes into one prime. This way you get the least multiple in terms of prime. Just simplify it to get the least common multiple.
By writing both of the above equations together, we get it as
$\begin{align}
  & 20=2\times 2\times 5 \\
 & 25=5\times 5 \\
\end{align}$
We can combine one 5 as common between both and write the least common multiple of 20, 25 in form of
$LCM\left( 20,25 \right)=5\times 5\times 2\times 2$
By simplifying above equation, we can write it in form of
LCM(20, 25) = 100
Therefore there must be a minimum of 100 students.

Note: Generally students confuse and think it is given either 20 or 25. So, it represents 2 different cases, do not combine them or else you will get the wrong answer. The idea of using the least common multiple here is very important. So, use it carefully in each and every step.