Question

By what number should 1365 be divided to get 31 as a quotient and 32 as remainder?

Answer 1

Hint: We have to apply the Euclid’s division lemma, $a=bq+r$ where $0 \le r < b$ that is Dividend = Divisor $\times $ Quotient + Remainder. The divisor is unknown, hence we will get the equation as $1365=31b+32$. Now, solve this equation to get the value of b, which is the divisor.

Complete step-by-step answer:
We are given that 1365 is the dividend, 31 is the quotient and 32 is the remainder.
Now, we have to find the divisor that is the number that divides 1365.
Here, we have to apply the Euclid’s division lemma. According to Euclid’s division lemma, if we have two positive integers a and b then there exists unique integers q and r which satisfy the condition $a=bq+r$ where $0\le r < b$.
Hence, by $a=bq+r$ we mean:
Dividend = Divisor $\times $ Quotient + Remainder
Here, we have $a = 1365, q = 31$ and $r = 32.$
Now, we have to find $b$.
By Euclid’s division lemma, we can write:

$ 1365=b\times 31+32 $

$ \Rightarrow 1365=31b+32 $

Next, by taking 32 to the left side, 32 becomes -32, so the equation,

$ \Rightarrow 1365-32=31b $

$ \Rightarrow 1333=31b $

In the next step, let’s shift 31 to the left-hand side of the equation. So, we get:
$ \Rightarrow \dfrac{1333}{31}=b $
$ \Rightarrow 43=b $
Hence, we can say that the divisor is 43.

Therefore, 1365 should be divided by 43 to get 31 as quotient and 32 as remainder.

Note: In Euclid’s division algorithm the remainder should be less than the divisor and also it should be greater than or equal to zero. Here, after getting the divisor, you can verify the answer by dividing the number 1365. If you are getting the quotient as 31 and remainder as 32, your answer is correct