a) Let a/b and c/d be two rational numbers. Prove that there exists an integer e such that (a/b) + (c/d) = e/lcm(b,d) (b) b)Show by example that if you write a/b + c/d = e/f where gcd(e; f) = 1, then f could be strictly smaller than lcm(b; d). Explain why this does not contradict part (a). (2 points) please explain

owsicag7

owsicag7

Answered question

2022-07-27

a) Let a/b and c/d be two rational numbers. Prove that there exists an integer e such that (a/b) + (c/d) = e/lcm(b,d) (b)
b)Show by example that if you write a/b + c/d = e/f where gcd(e; f) = 1, then f could be strictly smaller than lcm(b; d). Explain why this does not contradict part (a). (2 points) please explain

Answer & Explanation

abortargy

abortargy

Beginner2022-07-28Added 19 answers

a b + c d = a d + b c b d s i n c e   i f   t h e r e   a r e   t w o   i n t e g e r s   x . y t h e n x y = l e m ( x . y ) × g e d ( x . y ) t h e r e f o r e a d + b c b d = a d + b c l e m ( b . d ) g e d ( b . d ) = 1 l e m ( b . d ) ( a d g e d ( b . d ) + b c g e d ( b . d ) ) = 1 l e m ( b . d ) ( a k 1 + c k 2 ) t h e r e f o r e   e = a k 1 + c k 2 t h i s   c o m p l e t e s   t h e   p r o o f . a b + c d = e f a = 2 , b = 2 , c = 2 , d = 2 t h e n   a b + c d = 2 2 + 2 2 = 2 = 2 1 s o   w e   g e t   e = 2   a n d   f = 1. s u c h   t h a t   g e d ( e . f ) = 1

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