We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
+1
245
2
avatar

In how many different ways can 2/15 be represented as 1/A + 1/B, if A and B are positive integers with A less than or equal to B

 Jan 21, 2019
edited by Guest  Jan 21, 2019
 #1
avatar
+1

Is this what you mean?

 

1/8  + 1 / 120    = 2/15
1/9  +  1 / 45     = 2/15
1/10  +  1 / 30   = 2/15
1/12  +  1 / 20   = 2/15

1/15  +  1/15     = 2/15

 Jan 22, 2019
edited by Guest  Jan 22, 2019
 #2
avatar+23131 
+8

In how many different ways can 2/15 be represented as 1/A + 1/B,

if A and B are positive integers with A less than or equal to B

 

\(\text{For odd $n>2$ there is always at least one decomposition into exactly two unit fractions: $\dfrac{2}{n} = \dfrac{1}{A} + \dfrac{1}{B}$ } \\ \text{Finding all possibilities.}\\ \text{The prime factorization of $n^2$ results in all possible decompositions into two unit fractions.} \)

 

\(n=15\ \text{is odd} \\ n^2 =225 \\ \text{All divisors of $n^2=225$ are: $1,\ 3,\ 5,\ 9,\ 15,\ 25,\ 45,\ 75,\ 225$}\\ \text{Let $n^2=p\times q$ }\)

 

\(\begin{array}{|r|r|r|c|c|c|c|c|c|c|c|c| } \hline & p & q & n^2 & s & t & r & k & A & B & A\le B & \\ & & & =p\cdot q & = \frac{p+q}{2} & = \frac{p-q}{2} & =\frac{t}{2} & = \frac{15+\sqrt{15^2+t^2} }{2} & =k-r & =k+r & \\ \hline 1. & 225 & 1 & 225=225 \cdot 1 & 113 & 112 & 56 & 64 & 8 & 120 & \checkmark & \mathbf{\dfrac{2}{15} = \dfrac{1}{8} + \dfrac{1}{120}} \\ \hline 2. & 75 & 3 & 225= 73 \cdot 3 & 39 & 36 & 18 & 27 & 9 & 45 & \checkmark & \mathbf{\dfrac{2}{15} = \dfrac{1}{9} + \dfrac{1}{45}} \\ \hline 3. & 45 & 5 & 225= 45 \cdot 5 & 25 & 20 & 10 & 20 & 10 & 30 & \checkmark & \mathbf{\dfrac{2}{15} = \dfrac{1}{10} + \dfrac{1}{30}} \\ \hline 4. & 25 & 9 & 225= 25 \cdot 9 & 17 & 8 & 4 & 16 & 12 & 20 & \checkmark & \mathbf{\dfrac{2}{15} = \dfrac{1}{12} + \dfrac{1}{20}} \\ \hline 5. & 15 & 15 & 225= 15 \cdot 15 & 15 & 0 & 0 & 15 & 15 & 15 & \checkmark & \mathbf{\dfrac{2}{15} = \dfrac{1}{15} + \dfrac{1}{15}} \\ \hline \end{array}\)

 

laugh

 Jan 22, 2019
edited by heureka  Jan 22, 2019
edited by heureka  Jan 22, 2019

17 Online Users

avatar