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Let **be the binary operation on N give...

Let `**`be the binary operation on N given by `a**b = LCM`of a and b. Find
(i) `5 ** 7, 20 ** 16`
(ii) Is `*`commutative?
(iii) Is `**`associative?
(iv) Find the identity of `**`in N
(v) Which elements of N are invert

Text Solution

Verified by Experts

(i) `5***7=LCM of 5 and 7 =35`
`20***16=LCM of 20 and 16=80`
(ii)` ***` is commutative if
`a***b=b***a`
`a***b=LCM of a & b`
`b***a=LCM of b&a`
Since
`a***b=b***a AA a,b in N`
`***` is commutative.
(iii)`***` is associative if
`(a***b)***c=a***(b***c)`
`(a***b)***c `
`(LCM of a & b )***c`
`LCM of (LCM of a &b)&c`
`LCM of a,b &c `
`a***(b*c)`
` a***(LCM of b&c) `
`LCM of a&(LCM of b&c)`
`LCM of a,b &c`
`Since`
`(a***b)***c=a***(b***c)AA,b in N `
`*** is associative`
(iv) `e` is the identity of `***` if
`a***e=e***a=a`
`LCM of a & e=LCM of e&a=a`
Let `e=1`
`LCM of a&1=a`
`LCM of 1 &a =a`
`a***1=1***a=a AA in N `
Thus 1 is the identity of `***` in N.
(v) An element a in N is invertible if there is no element b in N such that `a***b=e=b***a`
Here b is the inverse of a
Here, `e=1`
`So, a***b=1=b***a`
`LCM of a&b =1=LCM of b&a`
Lcm of two numbers are 1 if both numbers are 1
So `a=b=1 `
Hence 1 is only invertible element in N for operation.
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