Question

Fill in the blanks.
When a whole number \[a\] is divided by a non-zero whole number \[b\], then there exist whole numbers \[q\] and \[r\] such that \[a = bq + r\], where either\[r = \] ______ or \[r = \]______.

Answer 1

Hint: Here, we will use the Euclid’s division lemma states that for any two positive integers \[a\] and \[b\] there exist two unique whole numbers \[q\] and \[r\] such that ,\[a = bq + r\] where \[0 \leqslant r < b\].

Complete step-by-step answer:
We are given that when a whole number \[a\] is divided by a non-zero whole number \[b\], then there exist whole numbers \[q\] and \[r\] such that \[a = bq + r\].
We know that Euclid’s division lemma tells us about the divisibility of integers. So we state that any positive integer \[a\] can be divided by any other positive integer \[b\] in such a way that it leaves a remainder \[r\].
Since we have that Euclid’s division lemma states that for any two positive integers \[a\] and \[b\] there exist two unique whole numbers \[q\] and \[r\] such that ,\[a = bq + r\] where \[0 \leqslant r < b\].
We know that the division algorithm, \[A = BQ + R\], where A is the dividend, B is the divisor, Q is the quotient and R is the remainder either \[R = 0\] or \[R < B\].
Hence, the values of \[r\] can be either \[r = 0\] or \[r < b\].

Note: While solving this question, be careful while comparing the given statement with Euclid’s division algorithm. One should know that an Euclid’s division lemma if we have two positive integers \[a\] and \[b\], then there exist unique integers \[q\] and \[r\] which satisfies the condition \[a = bq + r\], where \[0 \leqslant r < b\].