What is the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in form of a solid square?
a)900
b)400
c)1600
d)2500
Correct answer is option 'A'. Can you explain this answer?

Question

Answer

L.c.m of 12, 15, 18 & 20
12=2×2×3
15=3×5
20=2×2×5
here, L.c.m = 2×2×3×5×3.
the soldiers are in the form of a solid square.
hence, the required number of soliders
= 2×2×3×3×5×5
=900...

Answer

When u multiple the LCM, u won't get a perfect square which means

The product of LCM is 180 and 180 is not a perfect square. To get the perfect square multiply it with the natural number until u get the Perfect square. In this question 180*5= 900 so 5 is multiplied

Answer

To find the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in form of a solid square, we need to find the LCM (Least Common Multiple) of these numbers.

Method 1: Prime Factorization

12 = 2^2 x 3
15 = 3 x 5
18 = 2 x 3^2
20 = 2^2 x 5

The LCM can be found by taking the highest power of each prime factor that appears in the factorization of any of the numbers.

LCM = 2^2 x 3^2 x 5 = 180

So, the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in form of a solid square is 180^2 = 32,400.

Method 2: Division

We can also find the LCM by division method.

Step 1: Write the given numbers in a vertical column.

12 | 15 | 18 | 20
---|---|---|---
2 | 3 | 2 | 2
2 | 5 | 3 | 5
3 | | |

Step 2: Divide each number by its prime factors. Write down the prime factors and their highest powers on the top row.

Step 3: For each prime factor, take the highest power that appears in any column and write it in the bottom row.

2^2 x 3 x 5 = 60

Step 4: Multiply the prime factors in the bottom row to get the LCM.

LCM = 2^2 x 3 x 5 = 60

So, the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in form of a solid square is 60^2 = 3600.

However, this number is not a perfect square. We need to find the next perfect square that is greater than or equal to 3600.

40^2 = 1600

So, the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in form of a solid square is 40^2 x 9 = 36 x 1600 = 57,600.

Therefore, the correct answer is option A) 900, which is obtained by taking the square root of 57,600.

Answer

900

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