Number of pairs of coprime positive integers ( a , b ) such that a b + 14 b a is an integer?The problem I'm facing in this question is that is there any algebraic or short-cut method to find the pairs. Else I've done it by Hit and Trial but that consumes a lot of time.

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Number of pairs of coprime positive integers   (  a  ,  b  ) such that       a    b    +            14      b        a   is an integer?The problem I&#039;m facing in this question is that is there any algebraic or short-cut method to find the pairs. Else I&#039;ve done it by Hit and Trial but that consumes a lot of time.

Maeve Holloway · Accepted Answer

You are looking for coprime integers   a and   b for which      a    b    +  14      b    a    =                    a        2            +      14              b        2                    a      b        ,is an integer. Then   b divides       a    2  , and so   b  =  1 because   gcd  (  a  ,  b  )  =  1. Now the above reduces to                    a        2            +      14        a    ,which is an integer if and only if   a divides   14, i.e.   a  ∈  {  1  ,  2  ,  7  ,  14  }

Kendall Oneill · Accepted Answer

Let     x  =  a      /    b  .       x  +  14      /    x  =  n   so         x    2      −  n  x  +  14  =  0   so      b  ∣  1  ,    a  ∣  14   via Rational Root Test.

Number of pairs of coprime positive integers ( a , b ) such that a

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