Question

# A general can draw up his soldiers in the rows of 10, 15 and 18 soldiers and he also draws them up in the form of a solid square. Find the least number of soldiers with the general(a) 100(b) 3600(c) 900(d) 90

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Hint: To solve this question we will first of all find the LCM of 10, 15 and 18. LCM is the smallest positive integer that is divisible from all the given numbers. Then we will find the number of soldiers by making the LCM obtained as a perfect square. This is so as we need a solid square.

Given that the general draws the soldiers in rows of 10, 15 and 18.
We need to determine the total number of possible combinations using 10, 15 and 18.
This can be done by finding the LCM of 10, 15 & 18.
LCM $\to$ Least common multiple of two integers a & b is the smallest positive integer that is divisible by both a & b.
Denoted by LCM (a, b).
If we have three numbers then it is denoted by LCM (a, b, c).
Now we will proceed to determine LCM of 10, 15 & 18.
\begin{align} & 2\left| \!{\underline {\, 10,15,18 \,}} \right. \\ & 3\left| \!{\underline {\, 5,15,9 \,}} \right. \\ & 3\left| \!{\underline {\, 5,5,3 \,}} \right. \\ & 5\left| \!{\underline {\, 5,5,1 \,}} \right. \\ & \left| \!{\underline {\, 1,1,1 \,}} \right. \\ \end{align}
So the LCM of 10, 15 and 18 is $2\times 3\times 3\times 5=90$.
Now he wants to draw them up in the form of a solid square.
Then because LCM of 10, 15 and 18 is coming to be 90, then any multiple of 90 gives a solid square.
Let multiple be denoted by ‘k’.
Then, the number of soldiers to draw a solid square is 90k.
The least value of R for which 90k is a perfect square is 10.
Then, R = 10 $\Rightarrow$ the required number of soldiers is $90\times 10=900$.
So, the correct answer is “Option C”.

Note: The possibility of error in this question can be not assuming the multiple to be ‘k’ which would be wrong as we need to form a solid square. Hence we need to multiple to be assumed. Due to this misconception, the student might end up choosing option d as the right answer, so they must be very careful in reading the question and marking the right option.