Question

When we write the hcf of any integer in the form of linear combination, how it is written in this form3=75*13-81*12,while finding the hcf of 273&81

Answer 1

it is 237,not 273
every step is well explained

Between 81 and 237; 237 is greater than 81

Division lemma of 237 and 81:

Step 1: 237 = 81 × 2 + 75

Step 2: Since remainder 75 ≠ 0, division lemma is applied to 81 and 75 to get 81 = 75 × 1 + 6

Step 3: Since remainder 6 ≠ 0, division lemma is applied to 75 and 6 to get 75 = 6 × 12 + 3

Step 4: Since remainder 3 ≠ 0, division lemma is applied to 6 and 3 to get 6 = 3 × 2 + 0

The remainder is zero in step 4.

Therefore, the divisor i.e. 3 in this step is the H.C.F. of the given numbers.

The H.C.F. of 237 and 81 is 3

Step 5: From Step 3: 3 = 75 – 6 × 12 -----

From Step 2: 6 = 81 – 75 × 1

Thus, from Step 5, we, get 3 = 75 – (81 – 75 × 1) × 12

⇒ 3 = 75 – (81× 12 – 75 × 12)

Step 6 ⇒ 3 = 75 × 13 – 81× 12

From Step 1, 75 = 237 – 81 × 2

Thus, from Step 6;

3 = (237 – 81 × 2) × 13 – 81× 12
⇒ 3 = (237 × 13 – 81 × 26) – 81× 12
⇒ 3 = 237 × 13 – 81 × 38
⇒ H.C.F. of 237 and 81 = 237 × 13 + 81 × (–38)

237 × 13 + 81 × (–38) is the representation of H.C.F. of 237 and 81 as linear combination of 237 and 81.