Question

what is Euclid's division Algorithm ?

Answer 1

An algorithm means a series of well defined steps which provide a procedure of calculation repeated successively on the results of earlier steps till the desired result is obtained.

Theorem : If a and b are positive integers such that a = bq + r, then every common divisor of a and b is a common divisor of b and r, and vice-versa.
Proof : Let c be a common divisor of a and b. Then,
c| a ⇒ a = cq1 for some integer q1
c| b ⇒ b = cq2 for some integer q2.
Now, a = bq + r
⇒ r = a – bq
⇒ r = cq1 – cq2 q
⇒ r = c( q1 – q2q)
⇒ c | r
⇒ c| r and c | b
⇒ c is a common divisor of b and r.
Hence, a common divisor of a and b is a common divisor of b and r.
Euclids Division Algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.
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Let us state Euclid’s division algorithm clearly.
To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.
Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

This algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d, etc.

Answer 2

Euclid's division Algorithm is a method which make the HCF theory easy.
this method is 'bq+r'.