Question

State "Euclid's division Lemma".

Answer 1

Hint: Euclid's division Lemma( a proven statement) states that for two positive integers a and b, there exists unique integers q and r which satisfies the condition;
$a = bq + r$ where $0 \leqslant r \leqslant b$.
Using the above hint we will elaborate the solution further.

Complete step-by-step solution:
The statement for the Euclid's division lemma;
"For two positive integers a and b there exists unique integers q and r such that a = bq + r, where q is the quotient, b is the divisor, a is dividend and r is the remainder.
In more detailed manner, we can explain the Euclid's division Lemma as;
Given integers a and b with b not equal to 0 there exist unique integers q and r satisfying a = qb + r, the Lemma is valid for all a and b such that b not equal to zero, q and r are unique, r can be zero or greater than zero but must be less than absolute value of b, b can be bigger than a . If we divide any number using Euclid's division Lemma by two and get the remainder as zero then the number is even and if the remainder obtained is more than zero than the number or integer is odd.

Note: To calculate the highest common factor (HCF) of two positive integers x and y, Euclid's division algorithm is used. Euclid's division Lemma is generally used to prove other algorithms; such as Euclid's division lemma is used for proving positive odd integers of the form 2q +1.