find-integers-a-gt-b-gt-c-gt-0-such-that-1-a-2-b-3-c-1-

Question Number 181837 by mr W last updated on 01/Dec/22

f i n d i n t e g e r s a > b > c > 0 s u c h t h a t

\frac{1}{a} + \frac{2}{b} + \frac{3}{c} = 1

Commented by SEKRET last updated on 01/Dec/22

(2; 5; 30) (2; 6; 18) (2; 7; 14) (2; 8; 12)

(2; 10; 10) (2; 12; 9) (2; 16; 8) (2; 28; 7)

(3; 4; 18) (3; 6; 9) (3; 12; 6) (3; 30; 5)

(4; 3; 36) (4; 4; 12) (4; 8; 6) (5; 4; 10)

(5; 10; 5) (5; 40; 4) (15; 6; 5) (20; 10; 4)

(30; 3; 10) (36; 9; 4) (6; 3; 18) (6; 4; 9)

(6; 6; 6) (6; 24; 4) (8; 4; 8) (8; 16; 4) (10; 5; 6)

(12; 3; 2) (12; 12; 4) (14; 4; 7) \dots . .

Commented by mr W last updated on 01/Dec/22

o n l y s o l u t i o n s w i t h a > b > c > 0 a r e

r e q u e s t e d .

Answered by prakash jain last updated on 01/Dec/22

a = 2

b = 8

c = 12

Commented by mr W last updated on 01/Dec/22

t h a n k s f o r t r y i n g s i r!

b u t o n l y s o l u t i o n s w i t h a > b > c > 0 a r e

r e q u e s t e d .

Answered by MJS_new last updated on 01/Dec/22

a > b > c > 0 \Rightarrow c > 3

a = \frac{b c}{b c - 3 b - 2 c}

a > b \Rightarrow \frac{c}{b c - 3 b - 2 c} > 1 \land c > 3

\Leftrightarrow

c < b < \frac{3 c}{c - 3} \land c > 3 \Rightarrow c = 4 \land 4 < b < 12 \lor c = 5 \land 5 < b < 7.5

not many possibilities to try out

\Rightarrow

a = 36 \land b = 9 \land c = 4

a = 20 \land b = 10 \land c = 4

a = 15 \land b = 6 \land c = 5

are the only solutions

Commented by mr W last updated on 02/Dec/22

t h a n k s a l o t s i r!

Answered by mr W last updated on 02/Dec/22

a > b > c > 0

\Rightarrow \frac{1}{a} < \frac{1}{b} < \frac{1}{c}

1 = \frac{1}{a} + \frac{2}{b} + \frac{3}{c} < \frac{1}{c} + \frac{2}{c} + \frac{3}{c} = \frac{6}{c}

\Rightarrow c < 6

\frac{3}{c} = 1 - \frac{1}{a} - \frac{2}{b} < 1

\Rightarrow c > 3

\Rightarrow c m a y o n l y b e 4 o r 5 .

\underset{―}{w i t h c = 4 :}

\frac{1}{a} + \frac{2}{b} = 1 - \frac{3}{4} = \frac{1}{4}

\frac{1}{4} = \frac{1}{a} + \frac{2}{b} < \frac{1}{b} + \frac{2}{b} = \frac{3}{b}

\Rightarrow b < 12

\frac{1}{4} = \frac{1}{a} + \frac{2}{b} > \frac{2}{b}

\Rightarrow b > 8

\Rightarrow b m a y o n l y b e 9, 10 o r 11 .

b = 9 : \frac{1}{a} = \frac{1}{4} - \frac{2}{9} = \frac{1}{36} \Rightarrow a = 36 ✓

b = 10 : \frac{1}{a} = \frac{1}{4} - \frac{2}{10} = \frac{1}{20} \Rightarrow a = 20 ✓

b = 11 : \frac{1}{a} = \frac{1}{4} - \frac{2}{11} = \frac{7}{44} \Rightarrow a = \frac{44}{7} \times

\underset{―}{w i t h c = 5 :}

\frac{1}{a} + \frac{2}{b} = 1 - \frac{3}{5} = \frac{2}{5}

\frac{2}{5} = \frac{1}{a} + \frac{2}{b} < \frac{1}{b} + \frac{2}{b} = \frac{3}{b}

\Rightarrow b < \frac{15}{2} = 7.5

\frac{2}{5} = \frac{1}{a} + \frac{2}{b} > \frac{2}{b}

\Rightarrow b > 5

\Rightarrow b m a y o n l y b e 6 o r 7 .

b = 6 : \frac{1}{a} = \frac{2}{5} - \frac{2}{6} = \frac{1}{15} \Rightarrow a = 15 ✓

b = 7 : \frac{1}{a} = \frac{2}{5} - \frac{2}{7} = \frac{4}{35} \Rightarrow a = \frac{35}{4} \times

s u m m a r y :

(a, b, c) = (36, 9, 4), (20, 10, 4), (15, 6, 5)

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