Let-by-a-1-a-2-a-n-we-mean-LCM-of-a-1-a-2-a-n-where-a-i-N-Prove-or-disprove-that-a-b-b-c-a-b-c-

Question Number 8846 by Rasheed Soomro last updated on 31/Oct/16

Let by (a_{1}, a_{2}, \dots a_{n}) we mean LCM

of a_{1}, a_{2}, \dots a_{n}, where a_{i} \in N .

Prove or disprove that ((a, b), (b, c)) = (a, b, c) .

Answered by 123456 last updated on 01/Nov/16

lets say

x = (a, b)

y = (b, c)

z = (a, b, c)

by definition

x = (a, b) \Rightarrow a ∣ x \land b ∣ x

y = (b, c) \Rightarrow b ∣ y \land c ∣ y

z = (a, b, c) \Rightarrow a ∣ z \land b ∣ z \land c ∣ z

call u = (x, y)

u = (x, y) \Rightarrow x ∣ u \land y ∣ u

since a ∣ x and x ∣ u, a ∣ u

by same reason

b ∣ x \land x ∣ u \Rightarrow b ∣ u

b ∣ y \land y ∣ u \Rightarrow b ∣ u

c ∣ y \land y ∣ u \Rightarrow c ∣ u

so a ∣ u \land b ∣ u \land c ∣ u

the \min value that hold this is z

so z = u or

(a, b, c) = ((a, b), (b, c))

Commented by Rasheed Soomro last updated on 01/Nov/16

N i c e!