Let * be a binary operation on N given by a *b = 1 cm of a and b. Find the value of
20 * 16.

Is * (i) commutative, (ii) associative?


To find: LCM of 20 and 16


Prime factorizing 20 and 16 we get.


20 = 22 × 5


16 = 24


LCM of 20 and 16 = 24 × 5 = 80


(i) To find LCM highest power of each prime factor has been taken from both the numbers and multiplied.


So it is irrelevant in which order the number are taken as their prime factors will remain the same.


So LCM(a,b) = LCM(b,a)


So * is commutative


(ii) Let us assume that * is associative


LCM[LCM(a,b),c] = LCM[a,LCM(b,c)]


Let the prime factors of a be p1,p2


Let the prime factors of b be p2,p3


Let the prime factors of c be p3,p4


Let the higher factor of pi be qi for i = 1,2,3,4


LCM (a,b) = p1q1p2q2p3q3


LCM[LCM(a,b),c] = p1q1p2q2p3q3p4q4


LCM (b,c) = p2q2p3q3p4q4


LCM[a,LCM(b,c)] = p1q1p2q2p3q3p4q4


* is associative


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