Use Euclid's division algorithm to find the HCF of (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Explanation

(i) Here , `225 gt 135`
divide 225 by 135 to get 1 as quotient and 90 as remainder .
`therefore " " 225 = 135 xx 1 + 90`
Now the remainder is 90 which is not equal to zero
Again , divide 135 by 90 to get 1 as quotient and 45 as remainder
` therefore " " 135 = 90 xx 1 + 45`
Again divide by 90 by 45 to get 2 as quotient and zero as remainder
`therefore " " 90 = 45 xx 2 + 0`
Alternative method : It can also be shown in the following way

`therefore " " 225 = 135 xx 1 + 90`
`135 = 90 xx 1 + 45`
`90 = 45 xx 2 + 0`
Since , remainder = 0 `implies " "` recent divisor in the H.C.F.
`therefore` H.C.F. (225, 135) = 45
(ii) Here , `38220 gt 196`

`therefore " " 38220 = 196 xx 195 + 0`
`because` the remainder is zero `implies` recent divisor is the H.C.F.
`therefore` H.C.F. (38220 , 196) = 196
(iii) Here , `867 gt 255`
`therefore`

`therefore " " 867 = 255 xx 3 + 102 `
`255 = 102 xx 2 + 51`
`102 = 51 xx 2 + 0 `
Since remainder ` = 0 rArr` recent divisor is the `H.C.F.`
`:.` H.C.F. `(867,255) = 51`