Find-all-possible-solutions-1-s-1-t-1-u-1-v-1-With-s-t-u-v-N-and-s-lt-t-lt-u-lt-v-

Question Number 194158 by Frix last updated on 28/Jun/23

Find all possible solutions :

\frac{1}{s} + \frac{1}{t} + \frac{1}{u} + \frac{1}{v} = 1

With s, t, u, v \in N and s < t < u < v

Answered by AST last updated on 29/Jun/23

s < t \Rightarrow \frac{1}{s} > \frac{1}{t} \Rightarrow \frac{4}{s} > \frac{1}{s} + \frac{1}{t} + \frac{1}{u} + \frac{1}{v} = 1 \Rightarrow s < 4

s = 1 g i v e s a b s u r d r e s u l t, s o s = 2 o r 3

w h e n s = 2, \frac{1}{t} + \frac{1}{u} + \frac{1}{v} = \frac{1}{2} \Rightarrow \frac{3}{t} > \frac{1}{2} \Rightarrow t < 6

t = 1, 2 g i v e s a b s u r d r e s u l t s . . S o c o n s i d e r 3, 4 o r 5

t = 3 \Rightarrow \frac{1}{u} + \frac{1}{v} = \frac{1}{6} \Rightarrow u = \frac{6 (v - 6) + 36}{v - 6} = 6 + \frac{36}{v - 6} < v

\Rightarrow 6 v < v^{2} - 6 v \Rightarrow v^{2} - 12 v > 0 \Rightarrow v > 12

\Rightarrow v - 6 ∣ 36 \Rightarrow v = 15, 18, 24, 42 (s i n c e v > 12)

F o r e a c h o f t h e v a l u e s, w e g e t a n i n t e g e r v a l u e

f o r u = 10, 9, 8, 7 c o n s e c u t i v e l y

S i m i l a r l y, w e c a n g e t v a l u e s f o r t = 4 o r 5

t = 4 \Rightarrow \frac{2}{u} > \frac{1}{4} \Rightarrow u < 8 . . C h e c k i n g f o r 5, 6, 7

w e g e t u, v = (5, 20); (6, 12)

t = 5 \Rightarrow \frac{2}{u} > \frac{3}{10} \Rightarrow u < \frac{20}{3} \Rightarrow u = 6 g i v e s v = \frac{15}{2} \notin N

S o, c o n s i d e r s = 3; w e g e t \frac{1}{t} + \frac{1}{u} + \frac{1}{v} = \frac{2}{3}

\frac{3}{t} > \frac{2}{3} \Rightarrow t < 4.5 \Rightarrow t = 4 \Rightarrow \frac{1}{u} + \frac{1}{v} = \frac{5}{12}

\frac{2}{u} > \frac{5}{12} \Rightarrow u < 4.8 \Rightarrow n o s o l u t i o n s i n c e u > t = 4

s = 2 \Rightarrow t = 3 \Rightarrow (u, v) = (10, 15), (9, 18), (8, 24), (7, 42)

s = 2 \Rightarrow t = 4 \Rightarrow (u, v) = (5, 20); (6, 12)

Commented by Frix last updated on 29/Jun/23

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