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It is known that $2726,4472,5054,6412$ have the same remainder when they are divided by some two digit natural number $m$. Find the value of $m$.

Last updated date: 21st Jul 2024
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Hint: In order to find the natural number that divides all the numbers but leaves the same remainder, we must apply Euclid’s division algorithm which is $a=bq+r$. Then we have to apply this algorithm to all of the numbers and then we obtain the respective equations. Upon solving them, we obtained the required natural number.

Complete step-by-step solution:
Now let us have a brief regarding the Euclid division algorithm. It is also called as Euclid division Lemma which states that $a,b$ are positive integers, then there exists unique integers satisfying $q,r$ satisfying $a=bq+r$ where $0\le r< b$.
Now let us find the natural number $m$.
We know that Euclid division algorithm i.e. $a=bq+r$
By applying the division algorithm to the numbers, we get
\begin{align} & 2726=bx+r\to \left( 1 \right) \\ & 4472=ax+r\to \left( 2 \right) \\ & 5054=cx+r\to \left( 3 \right) \\ & 6412=dx+r\to \left( 4 \right) \\ \end{align}
Now we should subtract equation $\left( 1 \right)$ from $\left( 2 \right)$,$\left( 2 \right)$ from $\left( 3 \right)$, $\left( 3 \right)$ from $\left( 4 \right)$ and $\left( 4 \right)$ from $\left( 1 \right)$.
Upon subtracting, we get the following equations.
\begin{align} & 1746=\left( a-b \right)x \\ & 582=\left( c-a \right)x \\ & 1358=\left( d-c \right)x \\ & 3686=\left( d-a \right)x \\ \end{align}
We can express these equations numerically in the following way-
\begin{align} & \Rightarrow 1746=2\times 3\times 3\times 97 \\ & \Rightarrow 582=2\times 3\times 97 \\ & \Rightarrow 1358=2\times 7\times 97 \\ & \Rightarrow 3686=2\times 19\times 97 \\ \end{align}
So from the above expansion, we can observe that $97$is the only two digit number which is in common for all four numbers.
$\therefore$ The value of $m$is $97$.

Note: Using the Euclid division algorithm we can also find the HCF of the numbers. We must have a point to note that the numbers must be positive in order to apply the Euclid division algorithm in order to obtain a unique quotient and remainder.