Question

Express the hcf of 468 and 222 as 468x+222y where x,y are integer in two different ways

Answer 1

Answer:

The hcf is 6.

Two solutions for 468x + 222y = 6 are then:

x = -9 and y = 19

x = 28 and y = -59

Step-by-step explanation:

Use Euclid's Algorithm:

468 = 2 × 222 + 24

222 = 9 × 24 + 6

24 = 4 × 6 + 0

The hcf is 6.  Going backwards...

6 = 222 - 9 × 24

  = 222 - 9 × ( 468 - 2 × 222 )

  = 222 - 9 × 468 + 18 × 222

  = 19 × 222 - 9 × 468

So one solution already is

6 = 468x + 222y  with  x = -9 and y = 19

All other solutions are obtained from this one by adding / subtracting amounts that don't change the total of 6.

First, 222/6 = 37 and 468/6 = 78.

So...

6 = 468×( -9 + 37k ) + 222×( 19 - 78k )

To get other solutions, we just put in different values of k.

For instance, if we put k = 1, this results in

6 = 468x + 222y  with x = -9 + 37 = 28  and  y = 19 - 78 = -59