gcd-a-b-lcm-a-b-111-find-a-and-b-

Question Number 123304 by mordo last updated on 24/Nov/20

g c d (a; b) + l c m (a; b) = 111

f i n d a a n d b

Commented by mr W last updated on 24/Nov/20

a = 1

b = 110

Answered by MJS_new last updated on 24/Nov/20

(1) \gcd (a, b) = 1 \Rightarrow lcm (a, b) = a b = 110

(2) \gcd (a, b) = k \neq 1 \Rightarrow a = k x \land b = k y \Rightarrow

\Rightarrow lcm (a, b) = k x y \Rightarrow k + k x y = 111 \Leftrightarrow

\Leftrightarrow k = \frac{111}{x y + 1} with \gcd (x, y) = 1

in both cases we can try to find solutions

for a < b I get these pairs :

a b

1 110

2 55

3 108

5 22

10 11

12 27

37 74

Answered by floor(10²Eta[1]) last updated on 24/Nov/20

let d = \gcd (a, b) \Rightarrow a = dx, b = dy

where \gcd (x, y) = 1

d + dxy = 111 ∴ d (1 + xy) = 111

d ∣ 111 \Rightarrow d = 1, 3, 37, 111

Case 1 : d = 1 \Rightarrow xy = ab = 110 = 11.5.2

a = 11.5.2, b = 1

a = 11.5, b = 2

a = 11.2, b = 5

a = 11, b = 2.5

Case 2 : d = 3 \Rightarrow xy = 36 = 2^{2} . 3^{2}

x = 2^{2} . 3^{2}, y = 1 \Rightarrow a = 2^{2} . 3^{3}, b = 3

x = 3^{2}, y = 2^{2} \Rightarrow a = 3^{3}, b = 2^{2} . 3

Case 3 : d = 37 \Rightarrow xy = 2

x = 2, y = 1 \Rightarrow a = 37.2, b = 37

Case 4 : d = 111 (no sol .)

(a, b) = {(110, 1), (55, 2), (22, 5), (11, 10)

(108, 3), (27, 12), (74, 37)}

and the (b, a) pairs

Commented by mordo last updated on 25/Nov/20

t h a n k s

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