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-

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 \dots .