Question

3
Euclid's division algorithm is a
repeated application of division
lemma until we get remainder as
A)1
B)
C)infinitive
D)none of these​

Answer 1

Step-by-step explanation:

 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.

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Examples :

1) Use Euclid’s algorithm to find the 420 and 130.

Solution :

Step:1 Since 420 > 130 we apply the division lemma to 420 and 130 to get ,

420 = 130 x 3 + 30

Step:2 Since 30 ≠ 0 , we apply the division lemma to 130 and 30 to get

130 = 30 x 4 + 10

Step:3 Since 10 ≠ 0 , we apply the division lemma to 30 and 10 to get

30 = 10 x 3 + 0

The remainder has now become zero, so our procedure stops. Since the divisor at this Step is 10, the HCF of 420 and 130 is 10.

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2) Use Euclid’s algorithm to find the 65 and 117.

Solution :

Step:1 Since 117 > 65 we apply the division lemma to 117 and 65 to get ,

117 = 65 x 1 + 52

Step:2 Since 52 ≠ 0 , we apply the division lemma to 65 and 52 to get

65 = 52 x 1 + 13

Step:3 Since 13 ≠ 0 , we apply the division lemma to 52 and 13 to get

52 = 13 x 4 + 0

The remainder has now become zero, so our procedure stops. Since the divisor at this Step is 13, the HCF of 117 and 52 is 13.

Euclid's Geometry

• Euclid Geometry

• Euclids division lemma

• Euclids division Algorithm

• Fundamental Theorem of Arithmetic

• Finding HCF LCM of positive integers

• Proving Irrationality of Numbers

• Decimal expansion of Rational numbers

From Euclid Geometry to Real numbers