Question

Determine whether the equation 486 + 222 = 6 has a solution such that , ∈ If yes, find x and y. If not, explain your answer.

Answer 1

Answer:

468 = 2 x 2 x 3 x 3 x 13

222 = 2 x 3 x 37

HCF = 2 x 3 = 6

6 = 222 - 216

= 222 - (24) 9

= 222 - (468–444)9

= 222 - (468)9 + (444)9

= 222 - (468)9 + (222)(2)(9)

= 222 (19) - 468 (9)

= 468(-9) + 222(19)

6 = 468 x + 222 y …………(1)

The above is one way of expressing the HCF 6 of 468 and 222 where x = - 9 and y = 19

Now 6 = 468 (-9) + 222 (19) can be written as 468 (-9) + 222(19) + 468(222)-222(468)

= 468(213) + 222(-449)

= 468 X + 222 Y………(2)

Thus the HCF 6 has been expressed as

468 (-9) +222(19) as also 468(213)+222(-449) in two different ways.

Euclidian algorithm gives both the gcd and the integer linear combination.

468=2⋅222+24

222=9⋅24+6

24=4⋅6+0.

Therefore, gcd(468,222)=6 .

Now do the operations in reversed order,

6=222−9⋅24

=222−9⋅(468−2⋅222)

=19⋅222−9⋅468.

The question ask for two different ways to write it. We can give infinite of them. Notice that the least common multiple of 222 and 468 is

222⋅4686=17316,

therefore

6=19⋅222−9⋅468

=19⋅222−9⋅468+17316k−17316k

=19⋅222−9⋅468+78⋅222k−468⋅37k

=(19+78k)⋅222−(9+37k)⋅468.

For example, if k=1 we have

6=97⋅222–46⋅468,

and a new solution for each integer value of k .