Question

The least positive remainder when \[123 \times 125 \times 127\] is divided by 124 is

Answer 1

Hint: The question is related to the division of a polynomial. First we need to multiply the given three numbers and then we are dividing the final product by the number 124. By using the normal division we are going to obtain a required solution for the given question.

Complete step-by-step answer:
Here we have 2 methods to solve the given question. One is by a normal multiplication and division and the other one is by modulo.
Method 1:
Here we solve this question by normal multiplication and normal division.
Now consider \[123 \times 125 \times 127\], first we multiply the two terms i.e., 123 and 125 so, we have
\[ \Rightarrow 15375 \times 127\]
On multiplying 15375 and 127 we have
\[ \Rightarrow 1952625\]
Therefore we have
\[ \Rightarrow 123 \times 125 \times 127 = 1952625\]
Now we have to divide the number 1952625 by 124.
On dividing we have
\[124\overset{15746}{\overline{\left){\begin{align}
  & {1952625} \\
 & \underline{124} \\
 & 712 \\
 & \underline{620} \\
 & 926 \\
 & \underline{868} \\
 & 582 \\
& \underline{496} \\
 & 865 \\
 & \underline{744} \\
 & 121 \\
\end{align}}\right.}}\]
On dividing we got the remainder as 121.
Method 2:
If a and b are integers and \[n > 0\] we write \[a \equiv b\,\bmod \,n\] to mean \[n|(b - a)\] We read this as “a is congruent to b modulo (or mod) n.
Now consider the given question, the numbers \[123,125\,and\,127\]
We write these numbers in the form of modulo. so we have
The number 123 is written as
\[ \Rightarrow 123 \equiv - 1(\bmod \,124)\]
The number 125 is written as
\[ \Rightarrow 125 \equiv 1(\bmod \,124)\]
The number 127 is written as
\[ \Rightarrow 127 \equiv 3(\bmod \,124)\]
Therefore now we have
\[ \Rightarrow 123 \times 125 \times 127 \equiv ( - 1)(1)(3)(\bmod \,124)\]
On simplifying we have
\[ \Rightarrow 123 \times 125 \times 127 \equiv - 3(\bmod \,124)\]
The number is a negative so we add 124 so we have
\[ \Rightarrow 123 \times 125 \times 127 \equiv 124 - 3(\bmod \,124)\]
On simplifying we have
\[ \Rightarrow 123 \times 125 \times 127 \equiv 121(\bmod \,124)\]
Here the remainder is 121.
therefore the least positive remainder when \[123 \times 125 \times 127\] is divided by 124 is 121
So, the correct answer is “121”.

Note: To solve these kinds of problems the students must know about the simple arithmetic operations and tables of multiplication. The concept of congruence also helps to solve these kinds of problems. \[a \equiv b\,\bmod \,n\]implies \[n|(b - a)\], where (b-a) is divided by n. if we get the b term as negative we add the n to b, such that it will turn into positive.