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

Let `*`be the binary operation on N given by `a*b = LdotCdotMdot`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

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(i) 5 * 7 = L.C.M. of 5 and 7 = 35
20 * 16 = L.C.M. of 20 and 16 = 80
(ii) Let a,b, in N
`therefore` a* b = L.C.M. of a and b
= L.C.M. of b and a
=b*a
`therefore` Operation * is commutative .
(iii) Let a,b,c in N
`therefore ` a* ( b* c) = a* (L.C.M of b and c)
`" "= `L.C.M of a,b and c
`" "= `(L.C.M of a and b ) *c
`" "= `(a*b) *c
`therefore ` Operation * is assoicative.
(iv) We know that
L.C.M of a and 1 = .C.M of 1 and a=a
`therefore` In ,N identify element of operation * = 1
(v) Let `a in N` and `b in N ` be such that
`therefore " " a* b = b* a=1`
`rArr L.C.M "of" and b = 1`
Therefore, in N the element 1 is invertible with respect to operation *.
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