Question

express it as a linear combination of 81 and 237.
full explain this

Answer 1

Actually the question should be : express HCF of 81 and 237 as a linear combination of x and y.

ANSWER:
bY EUCLIDS division lemma

a = b* q+ r
⬇ ⬇ ⬇ ⬇
=>237= 81 * 2 + 75 ----(1)
=>81=75*1+6 ------(2)
=>75=6*12+3 -----(3)
=>6=3 *2 +0

HCF= 3

3=75 - 6*12 (FROM 3)
[6 = 81 - 75*1] (FROM 2)
=> 3 = 75 - (81-75*1) 12
=>3 = 75 - 81(12) + 75( 12)
=>3 = 75 (1+12) - 81 (12)
[75 = 237-81*2] (FROM 1)
=>3 = (237 -81 * 2)(13) - 81 (12)
=>3 = 237 (13) - 81 (26) - 81 (12)
=> 3 = 237 (13) +81 (-38)
=> 3 = 237 (x) + 81 (y). where x=13 and y = -38