Let ∗ be the binary operation on N given by a∗b=L.C.M. of a and b. Find (i) 5∗7 , 20∗16 (ii) is ∗ communitative? (iii) is ∗ associative? (iv)Find the identity of ∗ in N (v) which elements of N are invertible for the operation ∗?


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Question Number 93950 by oustmuchiya@gmail.com last updated on 16/May/20

$L e t * b e t h e b i n a r y o p e r a t i o n o n N$ $g i v e n b y a * b = L . C . M . o f a a n d b . F i n d$ $(i) 5 * 7, 20 * 16 (ii) is * c o m m u n i t a t i v e ?$ $(iii) is * a s s o c i a t i v e ?$ $(iv) F i n d t h e i d e n t i t y o f * i n N$ $(v) which elements of N a r e i n v e r t i b l e$ $f o r t h e o p e r a t i o n * ?$
	Answered by som(math1967) last updated on 16/May/20

	$i) 5 * 7 = L . C . M of 5, 7 = 35$ $20 * 16 = L . C . M of 20, 16 = 80$ $ii) L . C . M of a, b = L . C . M of$ $b, a$ $∴ * is commutative [for all a, b \in N]$ $iii) (L . C . M of a, b) * c$ $= L . C . M of a, b, c$ $= a * (L . C . M of b, c)$ $= L . C . M of a, b, c$ $∴ * is associative$
	Answered by som(math1967) last updated on 16/May/20

	$iv) Let e \in N be the identity$ $element in N$ $∴ a * e = a [a \in N]$ $L . C . M of a, e = a$ $but a \in N ∴ e = 1 is identity eliment$ $in N ans$
	Commented by oustmuchiya@gmail.com last updated on 16/May/20

	$W o w! A m i l l i o n t h a n k s i n d e e d P a p a .$
	Commented by oustmuchiya@gmail.com last updated on 16/May/20

	$H o w a b o u t t h i s o n e p l e a s e .$ $L e t ^{'} b e t h e b i n a r y o p e t a t i o n o n t h e$ $s e t {1, 2, 3, 4, 5} d e f i n e d b y a ^{'} b = H . C . F .$ $o f a and b . I s t h e o p e r a t i o n ^{'} s a m e$ $a s t h e o p e r a t i o n . j u s t i f y y o u r$ $a n s w e r .$
	Commented by som(math1967) last updated on 16/May/20

	$a * b = HCF of a, b [a, b \in {1, 2, 3, 4, 5}]$ $= b * a = HCF of b, a$ $so a * b commutative . . . .$
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