Exercise 1.2.28 (Solve $(a,b,c) = 10,\ [a,b,c] = 100$ )

Find all triples of positive integers a , b , c satisfying ( a , b , c ) = 10 and [ a , b , c ] = 100 simultaneously.

Answers

Proof. If a ∧ b ∧ c = 10 , there exist positive integers A , B , C such that

A = 10 a , B = 10 B , c = 10 C , A ∧ B ∧ C = 1 .

Then

a ∨ b ∨ c = 100 ⟺ 10 A ∨ 10 B ∨ 10 c = 100 ⟺ 10 ( A ∨ B ∨ C ) = 100 ⟺ A ∨ B ∨ C = 10 .

So the conditions a ∧ b ∧ c = 10 , a ∨ b ∨ c = 100 are equivalent to

A ∧ B ∧ C = 1 , A ∨ B ∨ C = 10 .

We obtain 36 ordered triplets ( A , B , C ) solutions, which give the 36 solutions ( a , b , c ) = ( 10 A , 10 B , 10 C ) :

( a , b , c ) ∈ { ( 10 , 10 , 100 ) , ( 10 , 20 , 50 ) , ( 10 , 20 , 100 ) , ( 10 , 50 , 20 ) , ( 10 , 50 , 100 ) , ( 10 , 100 , 10 ) , ( 10 , 100 , 20 ) , ( 10 , 100 , 50 ) , ( 10 , 100 , 100 ) , ( 20 , 10 , 50 ) , ( 20 , 10 , 100 ) , ( 20 , 20 , 50 ) , ( 20 , 50 , 10 ) , ( 20 , 50 , 20 ) , ( 20 , 50 , 50 ) , ( 20 , 50 , 100 ) , ( 20 , 100 , 10 ) , ( 20 , 100 , 50 ) , ( 50 , 10 , 20 ) , ( 50 , 10 , 100 ) , ( 50 , 20 , 10 ) , ( 50 , 20 , 20 ) , ( 50 , 20 , 50 ) , ( 50 , 20 , 100 ) , ( 50 , 50 , 20 ) , ( 50 , 100 , 10 ) , ( 50 , 100 , 20 ) , ( 100 , 10 , 10 ) , ( 100 , 10 , 20 ) , ( 100 , 10 , 50 ) , ( 100 , 10 , 100 ) , ( 100 , 20 , 10 ) , ( 100 , 20 , 50 ) , ( 100 , 50 , 10 ) , ( 100 , 50 , 20 ) , ( 100 , 100 , 10 ) }

(Thanks to Python!) □

User profile picture
2024-09-29 08:42
Comments