Question

Use Euclid's division algorithm to find the HCF of: (i) 135and225 (ii) 196and38220 (iii) 867and255

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Answer 1

(i). 135 and 225

Euclid's division lemma :

Let a and b be any two positive Integers .

Then there exist two unique whole numbers q and r such that

a = bq + r ,

0 ≤ r <b

Now ,

Clearly, 225 > 135

Start with a larger integer , that is 225.

Applying the Euclid's division lemma to 225 and 135, we get

225 = 135 x 1 + 90

Since the remainder 90 ≠ 0, we apply the Euclid's division lemma to divisor 135 and remainder 90 to get

135 = 90 x 1 + 45

We consider the new divisor 90 and remainder 45 and apply the division lemma to get

90 = 45 x 2 + 0

Now, the remainder at this stage is 0.

So, the divisor at this stage, ie, 45 is the HCF of 225 and 135.

(ii). 196 and 38220

Euclid's division lemma :

Let a and b be any two positive Integers .

Then there exist two unique whole numbers q and r such that

a = bq + r ,

0 ≤ r <b

Now ,

Clearly, 38220 > 196

Start with a larger integer , that is 38220.

Applying the Euclid's division lemma to 38220 and 196, we get

38220 = 196 x 195 + 0

Now, the remainder at this stage is 0.

So, the divisor at this stage, ie, 196 is the HCF of 38220 and 196.

(iii) . 867 and 255

Euclid's division lemma :

Let a and b be any two positive Integers .

Then there exist two unique whole numbers q and r such that

a = bq + r ,

0 ≤ r <b

Now ,

Clearly, 867 > 255

Start with a larger integer , that is 255.

Applying the Euclid's division lemma to 867 and 255, we get

867 = 255 x 3 + 102

Since the remainder 102 ≠ 0, we apply the Euclid's division lemma to divisor 255 and remainder 102 to get

255 = 102 x 2 + 51

We consider the new divisor 102 and remainder 51 and apply the division lemma to get

102 = 51 x 2 + 0

Now, the remainder at this stage is 0.

So, the divisor at this stage, ie, 51 is the HCF of 867 and 255.

Answer 2

(i) 135 and 225

Since 225 > 135, we apply the division lemma to 225 and 135 to obtain

225 = 135 × 1 + 90

Since remainder 90 ≠ 0, we apply the division lemma to 135 and 90 to obtain

135 = 90 × 1 + 45

We consider the new divisor 90 and new remainder 45, and apply the division lemma to obtain

90 = 2 × 45 + 0

Since the remainder is zero at this stage so our procedure stops.

∴The HCF of 135 and 225 is 45.

(ii) 196 and 38220

Since 38220 > 196, we apply the division lemma to 38220 and 196 to obtain

38220 = 196 × 195 + 0

Since ,the remainder at this stage is Zero so our procedure stops.

∴ The HCF of 196 and 38220 is 196.

(iii) 867 and 255

Since 867 > 255, we apply the division lemma to 867 and 255 to obtain

867 = 255 × 3 + 102

Since remainder 102 ≠ 0, we apply the division lemma to 255 and 102 to obtain

255 = 102 × 2 + 51

We consider the new divisor 102 and new remainder 51, and apply the division lemma to obtain

102 = 51 × 2 + 0

Since the remainder at this stage is zero so our procedure stops.

∴ The HCF of 867 and 225 is 51.