Question

Express the HCF of468 and 222 as 468x+222y where x,y are integers in two different ways

Answer 1

HCF of 468 and 222 is expressed as 468(-9) + 222(19) = 6(HCF of 222 ad 468).

• To solve this type of questions we use Euclid's algorithm to find out the solution

468 = (222 × 2) + 24    (we express the greatest number in terms of smaller number)

222 = (24 × 9) + 6  (Now we express the other number in terms of the previous remainder)

24 = (6 × 4) + 0 (Now we express the other number in terms of the previous remainder)

Since the remainder obtained is zero , we know that the HCF of 222 and 468 is 6.

• We will use this above equations to find our solution ,

Now we take

6 = 222 - (24 × 9)

6 = 222 - {(468 – 222 × 2) × 9 [where 468 = 222 × 2 + 24]

6 = 222 - {468 × 9 – 222 × 2 × 9}

6 = 222 - (468 × 9) + (222 × 18)

6 = 222 +(222 × 18) - (468 × 9)

6 = 222[1 + 18] – 468 x 9

6 = 222 × 19 – 468 × 9

6 = 468 × (-9) + 222 × 19

• Comparing the above equation with required one we get x=-9 and y=19

Answer 2

Step-by-step explanation:

Given Express the HCF of 468 and 222 as 468 x+222 y where x,y are integers in two different ways

• Using Euclid division we get
• 468 = (222 x 2) + 24
• 222 = (24 x 9) + 6
• 24 = (6 x 4) + 0
• Now since remainder is 0, hcf = 6
• So we get
• 6 = 222 – (24 x 9)
• 6 = 222 – {(468 – 222 x 2) x 9 (where 468 = 222 x 2 + 24)
• 6 = 222 – {(468 x 9 – 222 x 2 x 9}
• 6 = 222 – (468 x 9) + (222 x 18)
• 6 = 222 + (222 x 18) – (468 x 9)
• 6 = 222 (1 + 18) – 468 x 9
• 6 = 222 x 19 – 468 x 9
• 6 = 468 x (-9) + 222 x (19)  

Now 468 x (-9) + 222 x 19 is of the form 468 x + 222 y