Question

The least positive remainder, when $123\times 125\times 127$ is divided by $124$ is

• A. $4$
• B. $120$
• C. $121$
• D. $130$

Answer 1

The correct option is C

$121$

Explanation of the correct option:

We know that $123,125$ and $127$ can be written as:

$$\begin{gathered} 123=-1\left(\mathrm{mod}124\right) \\ 125=1\left(\mathrm{mod}124\right) \\ 127=3\left(\mathrm{mod}124\right) \end{gathered}$$

Thus,

$\begin{array}{rcl}123\times 125\times 127& =& \left(-1\right)\left(1\right)\left(3\right)\left(\mathrm{mod}124\right)\\ & =& -3\left(\mathrm{mod}124\right)\\ & =& 121\left(\mathrm{mod}124\right)\end{array}$

Therefore, when $123\times 125\times 127$ is divided by $124$, the remainder is $121$.

Hence option C is the correct option.