p-q-are-two-natural-number-and-p-6-2p-4-4p-2-p-9-8p-3-1-4q-5-6q-then-find-the-minimum-possible-value-of-p-q-

Question Number 27996 by JI Siam last updated on 18/Jan/18

p, q are two natural number and

\frac{p^{6} + {2 p}^{4} + {4 p}^{2}}{p^{9} - {8 p}^{3}} - \frac{1}{4 q} = \frac{5}{6 q},

then find the minimum

possible value of p + q

Answered by prakash jain last updated on 18/Jan/18

\frac{p^{6} + {2 p}^{4} + {4 p}^{2}}{p^{9} - {8 p}^{3}} - \frac{1}{4 q} = \frac{5}{6 q},

\frac{p^{2} (p^{4} + {2 p}^{2} + 4)}{p^{3} (p^{6} - 2^{3})} = \frac{5}{6 q} + \frac{1}{4 q}

\frac{(p^{4} + {2 p}^{2} + 4)}{p (p^{2} - 2) (p^{4} + 2 p^{2} + 4)} = \frac{13}{12 q}

q = \frac{13}{12} p (p^{2} - 2)

p (p^{2} - 2) is a multiple of 12

p = 6 q = \frac{13}{2} (6^{2} - 2)

Commented by prakash jain last updated on 18/Jan/18

p = 13, q = 12 (13^{2} - 2)

Commented by JI Siam last updated on 18/Jan/18

i think p = 12, q = 13 (12^{2} - 2) sir

Commented by prakash jain last updated on 18/Jan/18

thanks . corrected p = 6