Question

What is the smallest number of soldiers in a regiment that will allow the soldiers to be lined up in rows of exactly 8,12, or 18 and form a square each time?

Answer 1

$\\\large{\dag{\underline{\underline{\sf{\pmb{Question:}}}}}}$

What is the smallest number of soldiers in a regiment that will allow the soldiers to be lined up in rows of exactly 8, 12, or 18 and form a square each time?

$\\ \large{\dag{\underline{\underline{\sf{\pmb{Required \: Answer:}}}}}}$

$\\ \underline{\underline{\tt{Given:}}}$

• The lines of the soldiers are to be in a row of 8, 12, or 18.
• The lines are to be a square at the same time.

$\\ \underline{\underline{\tt{To \: Find:}}}$

• The smallest number of the soldiers in the regiment.

$\\ \underline{\underline{\tt{Solution:}}}$

Here we were given,

• The lines of the soldiers are to be in a row of 8, 12, or 18 and a square at the same time.

Therefore, the LCM (Least Common Multiple) of 8, 12 & 18 will be the smallest number of the soldiers.

Now,

$\\ \sf{Prime \: factors \: of \:8 = \bold{2 \times 2 \times 2}}$

$\sf{Prime \: factors \: of \: 12 = \bold{2 \times 2 \times 3}}$

$\sf{Prime \: factors \: of \: 18 = \bold{2 \times 3 \times 3}}$

$\\ \sf{\therefore{ The \: LCM \: of \: 8,12,18 = 2 \times 2 \times 2 \times 3 \times 3 = \bold{72}}}$

But since the soldiers are in the form of square, so LCM needs to be perfect square. Now we need to multiply it by 2 in order to make it a perfect square.

$\\ \sf{\therefore{The \: number \: of \: the \: soldiers =( 2 \times 2) \times (2 \times 2) \times (3 \times 3 )= \red{\bold{144}}}}$

Hence, the number of the soldiers will be 144