How many pairs?

Number Theory Level pending

How many pairs with positive integers ( a , b ) (a,b) where a + b ≤ 100 a+b\le 100 satisfy the following equation?

a + b − 1 a − 1 + b = 17 \large \frac{a + b^{-1}}{a^{-1}+b} =17

2 4 3 1 5 0

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Tom Engelsman
Mar 12, 2017

The LHS can be simplified into:

a + 1 / b 1 / a + b = a b + 1 b ⋅ a a b + 1 = a b = 17 \frac{a + 1/b}{1/a + b} = \frac{ab +1 }{b} \cdot \frac{a}{ab + 1} = \frac{a}{b} = 17

If a + b ≤ 100 a + b \le 100 for a , b ∈ N a,b \in \mathbb{N} ( a > b ) , (a > b), then the required ordered-pairs are:

( a , b ) = ( 17 , 1 ) ; ( 34 , 2 ) ; ( 51 , 3 ) ; ( 68 , 4 ) ; ( 85 , 5 ) (a,b) = (17,1); (34,2); (51,3); (68,4); (85,5)

hence there are five such solutions.

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Nice solution. Thank you.

Hana Wehbi - 4 years, 3 months ago

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No prob, Hana! Thanks :)

tom engelsman - 4 years, 3 months ago

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