Find a , b ∈ N such that 1 a + 1 b + 1 ( a , b ) + 1 [ a , b ] = 1 2 My try : [ a , b ] + ( a , b ) [ a , b ] ⋅ ( a , b ) = a b − 2 a − 2 b 2 a b .Now what to do?

Question

Find   a  ,  b  ∈      N   such that       1    a    +      1    b    +      1          (      a      ,      b      )        +      1          [      a      ,      b      ]        =      1    2  My try :            [      a      ,      b      ]      +      (      a      ,      b      )              [      a      ,      b      ]      ⋅      (      a      ,      b      )        =            a      b      −      2      a      −      2      b              2      a      b        .Now what to do?

Sanaa Holder · Accepted Answer

Given   (  c  ,  d  )  =  1 and   g, we get the pair   (  a  ,  b  )  =  (  g  c  ,  g  d  ) and                              1          2                                    =                  1          a                +                  1          b                +                  1                      (            a            ,            b            )                          +                  1                      [            a            ,            b            ]                                                            =                  1                      g            c                          +                  1                      g            d                          +                  1          g                +                  1                      g            c            d                                                            =                  1          g                          (          1          +                      1            c                    )                          (          1          +                      1            d                    )                    That is  g  =  2      (    1    +          1      c        )        (    1    +          1      d        )  Here are all the possibilities                    c                    d                    g                    (        g        c        ,        g        d        )                            1                    1                    8                    (        8        ,        8        )                            1                    2                    6                    (        6        ,        12        )                            1                    4                    5                    (        5        ,        20        )                            2                    3                    4                    (        8        ,        12        )                            3                    8                    3                    (        9        ,        24        )                            4                    5                    3                    (        12        ,        15        )

Find a,b in NN such that 1/a+1/b+1/(a,b)+1/[a,b]=1/2

Answered question

Answer & Explanation

New Questions in Pre-Algebra