A trade has $612$ dettol soaps and $342$ pear soaps. He packs them in boxes and each box contains exactly one type of soap. If every box contains the same number of soaps, then what will be the number of soaps in each box such that the number of boxes is the least?

Given:

No. of  dettol soaps $=612$

No. of  pear soaps $=342$

We have to find the number of soaps in each box so that the number of boxes is the least.

For this, we will find the H.C.F of  $612$ and $342$ using Euclid’s Division Lemma method.

According to Euclid Division Lemma, for any two positive integers $a \mathrm{and} b$, there exists unique integer $q \mathrm{and} r$, which satisfies the condition $a = bq + r$ where $0 \le r < b$. 

In $a = bq + r$, $a$ is the dividend, $b$ is the divisor, $q$ is the quotient and $r$ is the remainder.

Now, in $612 and 342, 612$ is the greatest, so $a=612 and b=314$.

So, we have

$\begin{array}{rcl}612& =& 342\times 1+270\end{array}$

Now the divisor $342$ will become the dividend and the remainder $270$ will become the divisor, and we get

$\begin{array}{rcl}342& =& 270\times 1+72\end{array}$

Similarly,

$\begin{array}{rcl}270& =& 72\times 3+54\\ 72& =& 54\times 1+18\\ 54& =& 18\times 3+0\end{array}$

Now, as the remainder is $0$, therefore, we will not proceed further and the HCF will be the divisor $b$ in the last step. So, the H.C.F is $18$.

Therefore, there should be $18$ soaps in each box so that the number of boxes is the least.

Hence, the correct answer is option C.