If d is the HCF of 40 and 65, find the value of the integers x and y which satisfy d=40x + 65y. Show how the values of x and y have been derived step by step.

Asked by Jerlin George | 28th Sep, 2015, 01:57: PM

Expert Answer:

F i r s t space l e t space u s space f i n d space t h e space H C F space o f space 40 space a n d space 65 65 equals 1 cross times 40 plus 25... left parenthesis 1 right parenthesis 40 equals 1 cross times 25 plus 15... left parenthesis 2 right parenthesis 25 equals 1 cross times 15 plus 10... left parenthesis 3 right parenthesis 15 equals 1 cross times 10 plus 5... left parenthesis 4 right parenthesis 10 equals 2 cross times 5 plus 0... left parenthesis 5 right parenthesis T h u s space H C F space o f space 40 space a n d space 65 space i s space 5. F r o m space e q u a t i o n space left parenthesis 4 right parenthesis comma space w e space h a v e comma 5 equals 15 minus 1 cross times 10 F r o m space e q u a t i o n space left parenthesis 3 right parenthesis comma space w e space h a v e comma 5 equals 15 minus 1 cross times open parentheses 25 minus 1 cross times 15 close parentheses space space equals 15 minus 25 plus 15 space space equals 2 cross times 15 minus 25 F r o m space e q u a t i o n space left parenthesis 2 right parenthesis comma space w e space h a v e comma 5 equals 2 cross times open parentheses 40 minus 1 cross times 25 close parentheses minus 25 space space equals 2 cross times 40 minus 2 cross times 25 minus 1 cross times 25 space space equals 2 cross times 40 minus 3 cross times 25 F r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma 5 equals 2 cross times 40 minus 3 cross times open parentheses 65 minus 1 cross times 40 close parentheses space space equals 2 cross times 40 minus 3 cross times 65 plus 3 cross times 40 space space equals 5 cross times 40 minus 3 cross times 65 space space equals 5 cross times 40 plus open parentheses negative 3 close parentheses cross times 65 space space equals 40 x plus 65 y comma space w h e r e space x equals 5 space a n d space y equals negative 3

Answered by Vimala Ramamurthy | 29th Sep, 2015, 09:14: AM