# find the numbers of boys

• Sep 19th 2013, 03:26 AM
Zshan96
find the numbers of boys
The boys of a school can be arranged in 12 15 and 18 equal rows and also into a solid square. What is the least number of boys that school can have? (Hint find the lcm)
and the answer is 900 but my lcm is 180 and 180 isn't a perfect squred? I know my probpem will be solved!(Happy)
• Sep 19th 2013, 05:17 AM
HallsofIvy
Re: find the numbers of boys
No, 180 is not a perfect square! Neither is 2(180)= 360 nor 3(180)= 540, nor 4(180)= 720 but 5(180)= 900 is!

The point of this problem is that because 180 is the least common multiple, any common multiple of the numbers must be a multiple of 180- and 900 is the smallestmultiple of 180 that is a perfect square.

A more efficient method of finding that would be to note that 180= 9(20)= 9(4)(5). 9 and 4 are already "perfect squares". To get a perfect square, you need to multiply by 5: 9(4)(25)= 900.
• Sep 19th 2013, 08:11 AM
Zshan96
Re: find the numbers of boys
Thanks hallsoflvy for helping me and i think first method is simplethanks again
• Sep 19th 2013, 03:34 PM
Soroban
Re: find the numbers of boys
Hello, Zshan96!

Quote:

The boys of a school can be arranged in 12, 15, and 18 equal rows and also into a solid square.
What is the least number of boys that school can have? .(Hint: find the lcm.)

And the answer is 900 but my lcm is 180, and 180 isn't a perfect square.

Look at it this way . . .

. . $\displaystyle \begin{array}{ccc} 12 &=& 2^2\cdot 3 \\ 15 &=& 3\cdot 5 \\ 18 &=& 2\cdot 3^2\end{array}$

Hence: .$\displaystyle \text{LCM}(12,15,18) \:=\:2^2\cdot3^2\cdot5$

But this is not a square.
A square has even exponents on each of its prime factors.
. . And $\displaystyle 5=5^1$ has an odd exponent.

Therefore, we need "another 5".
. . $\displaystyle N \;=\;2^3\cdot 3^2\cdot 5^{\color{red}2} \:=\:900$
• Sep 19th 2013, 07:06 PM
Zshan96
Re: find the numbers of boys
Thanks soroban but i was taking lcm by common devision thanks brothers
• Sep 19th 2013, 07:12 PM
Zshan96
Re: find the numbers of boys
Thanks soroban but i was taking lcm by common devision thanks brothers another thing i want to ask that my lcm is 180 but why we take lcm as exponent method