Question

If the HCF of 8️⃣1️⃣ and 2️⃣3️⃣7️⃣ is expressible in form of 8️⃣1️⃣ X + 2️⃣3️⃣7️⃣Y , find the value of “x” and “y” ❓




**•reward42 Pts instantly•**

Answer 1

What is Euclid's Division Algorithm?

$\bullet$ It is a technique to compute the Highest Common Factor (HCF) of two given positive integers.

Now:

By Euclid's Division Algorithm:

$\implies$ $\sf{237 = 81(2) + 75}$

$\implies$ $\sf{81 = 75(1) +(6)}$

$\implies$ $\sf{75 = 6(12) + (3)}$

$\implies$ $\sf{6 = 3(2) + (0)}$

Therefore: HCF = 3

Expressing it in the form of:

$\boxed{\sf{237\:y+81\:x=HCF}}$

From 2nd last step:

$\implies$ $\sf{3 = 75 - 6 (12)}$

Substituting:

3 = 75 - 6 (12)

3 = 75 - (81 - 75 × 1) 12

3 = 13 × 75 - 12 × 81

3 = 13 × (237 - 81 × 2) - 12 × 81

3 = 13 × 237 - 26 × 81 - 12 × 81

3 = 13 × 237 - 38 × 81

3 = 237 y + 81 x

We need an expression in the form:

$\boxed{\sf{237\:y + 81\:x}}$

That is:

$\sf{3 = 237(13) + 81(-38)}$

Therefore:

$\implies$ $\boxed{\huge{\sf{x = - 38}}}$

$\implies$ $\boxed{\huge{\sf{y = 13}}}$

Answer 2

Answer:

x = -38, y = 13

Step-by-step explanation:

Among 81 and 237; 237 > 81

(i)

Since 237 > 81, we apply the division lemma to 237 and 81

237 = 81 * 2 + 75

(ii) Remainder ≠ 0

Since 81 > 75, we apply the division lemma to 81 and 75

81 = 75 * 1 + 6

(iii) Remainder ≠ 0

Since 75 > 6, we apply the division lemma to 75 and 6

75 = 6 * 12 + 3

(iv) Remainder ≠ 0.

Since 6  > 3, we apply the division lemma to 6 and 3

6 = 3 * 2 + 0

In this step, remainder = 0.

$\textbf{\underline{\underline{Thus, HCF(237,81) = 3}}}$

Now,

From Step (iii), we get

3 = 75 - 6 * 12

  = 75 - (81 - 75 * 1) * 12

  = 75 - (81 * 12 - 75 * 1 * 12)

  = 75 - 81 * 12 + 75 * 12

  = 75(1 + 12) - 81 * 12

  = 75 * 13 - 81 * 12

  = 13(237 - 81 * 12) - 81 * 12

  = 13 * 237 - 81 * 2 * 13 - 81 * 12

   = 237 * 13 - 81(26 + 12)

   = 237 * 13 - 81 * 38

   = 81 * (-38) + 237 * (13)

In the above relationship the H.C.F of 81 and 237 are in the form of 81x and 237y,where x = -38,y = 13.

$\therefore \textbf{\underline{\underline{x = -38,\ y = 13}}}$

Hope it helps!