a-b-c-N-1-a-1-b-1-c-16-37-a-b-c-

Question Number 157701 by MathSh last updated on 26/Oct/21

a; b; c \in N

\frac{1}{a + \frac{1}{b + \frac{1}{c}}} = \frac{16}{37} \Rightarrow a + b + c = ?

Answered by mindispower last updated on 26/Oct/21

\frac{1}{2 + \frac{5}{16}} = \frac{1}{2 + \frac{1}{3 + \frac{1}{5}}}

Commented by MathSh last updated on 26/Oct/21

thank you so much dear S er very nice

Answered by Rasheed.Sindhi last updated on 26/Oct/21

\frac{1}{a + \frac{1}{b + \frac{1}{c}}} = \frac{16}{37}; a, b, c \in N \Rightarrow a + b + c = ?

\frac{1}{a + \frac{1}{\frac{bc + 1}{c}}} = \frac{16}{37}

\frac{1}{a + \frac{c}{bc + 1}} = \frac{16}{37}

\frac{1}{\frac{abc + a + c}{bc + 1}} = \frac{16}{37}

\frac{bc + 1}{a (bc + 1) + c} = \frac{16}{37}

37 (bc + 1) = 16 (a (bc + 1) + c)

16 ∣ (bc + 1) \Rightarrow bc + 1 = 16, 32, 48, \dots

bc + 1 = 16 \Rightarrow bc = 15 = 3 \times 5

37 (16) = 16 (16 a + 5)

16 a + 5 = 37 \Rightarrow a = 2

(a, b, c) = (2, 3, 5) (O n e o f t h e s o l u t i o n s)

I t h i n k n o o t h e r s o l u t i o n .

Commented by MathSh last updated on 26/Oct/21

thank you so much dear S er very nice

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