Question

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 $$m$$.

Answer 1

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
$$\begin{array}{rl}& 2726=bx+r\to \left(1\right)\\ & 4472=ax+r\to \left(2\right)\\ & 5054=cx+r\to \left(3\right)\\ & 6412=dx+r\to \left(4\right)\end{array}$$
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.
$$\begin{array}{rl}& 1746=(a-b)x\\ & 582=(c-a)x\\ & 1358=(d-c)x\\ & 3686=(d-a)x\end{array}$$
We can express these equations numerically in the following way-
$$\begin{array}{rl}& \Rightarrow 1746=2\times 3\times 3\times 97\\ & \Rightarrow 582=2\times 3\times 97\\ & \Rightarrow 1358=2\times 7\times 97\\ & \Rightarrow 3686=2\times 19\times 97\end{array}$$
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.