Question

If 39 is divided by 9 then there exists two other unique non – negative integers q and r, find the value of q and r?

Answer 1

Hint: Use the division algorithm which states that the dividend is equal to the divisor multiplied with the quotient and its sum is taken with the remainder, mathematically dividend = (divisor $\times $ quotient) + remainder. Now assume 39 as the dividend and 9 as the divisor and write it in the form 39 = (9 $\times $ q) + r, where q is the quotient and r is the remainder.

Complete step by step solution:
Here we have been asked to divide 39 by 9 and find the two non – negative integers that exist as q and r. First we need to understand what does q and r denote.
Now, the division algorithm states that if we divide a number (the dividend) by another number (the divisor) then we get two numbers called the quotient and the remainder that satisfies the relation dividend = (divisor $\times $ quotient) + remainder mathematically. The quotient and the remainder are non – negative integers.
Let us come to the question. we have to divide 39 by 9 that means 39 is the dividend and 9 is the divisor. We can say that q is the quotient and r is the remainder.
$\Rightarrow 39=9q+r..........\left( i \right)$
When we divide 39 by 9 we get,
$\Rightarrow 39=\left( 9\times 4 \right)+3........\left( ii \right)$
On comparing the relations (i) and (ii) we get,
$\Rightarrow $ q = 4 and r = 3
Hence the two non – negative integers are 4 and 3.

Note: Do not try to form linear equations in q and r using the two relations obtained because you will get only one relation and two variables which cannot be solved. So just compare the coefficients to get the answer. The above used division algorithm is called the Euclid’s division algorithm or Euclid’s division lemma.