If the HCF of 56 and 72 is `m` then find `x` and `y` such that `m=56x+72y`

Explanation

Applying Euclid's division lemma to 56 and 72 , we get
(i) `72 = 56 xx 1+ 16` Here , remainder `ne` 0
(ii) `56 = 16 xx 3 + 8` Here , remainder `ne 0`
(iii) `16 = 8 xx 2 + 0` Here , remainder = 0
`therefore` H.C.F. of 56 and 72 is 8.
Now , `" " 8 = 56 - 16 xx 3`
= `56 - (72 - 56 xx1) xx 3`
= `56 - 72 xx 3 + 56 xx 3`
`= 56 xx 4 + 72 xx (-3)`
which is of the form 56x + 72y
`therefore " " x=4` and y = `-3`
But we can express equation (1) further as
`8 = 56 xx 4 + 72 xx (-3)`
`= 56 xx 4 - underset("same number")underset("Substract and add")ubrace(56 xx 72 + 56 xx 72) + 72 xx (-3)`
= ` 56(4 + 72) + 72(-56-3)`
=`56 xx 76 + 72 xx (-59)`
which is of the form 56x + 72y
`therefore " "` x = 76 , y = `-59`
Also , we can write
` 8 = 56 xx 4 + 72 xx (-3)` as below
`8 = 56 xx 4 + 72 xx (-3)` as below
`8 = 56 xx 4 - underset("same number")underset("Subtract and add")ubrace(56 xx 72 + 56 xx 72) + 72 xx (-3)`
`implies " " = 56( 4-72) + 72 (56-3)`
= `56 xx (-68) + 72 xx 53`
which is of the form equation 56x + 72y
`therefore " " x = -68`, y = 53
Hence , from equation (1) , (2) and (3) , we can say that x and y are not unique.