+166 Suppose that p is prime and \(1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p\). How many possible values of p are there?
+26367 Suppose that p is prime and
\(1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p. \)
1007_p+306_p+113_p+125_p+6_p=142_p+271_p+360_p.
How many possible values of p are there?
\(\small{ \begin{array}{|rcll|} \hline 1007_p+306_p+113_p+125_p+6_p &=& 142_p+271_p+360_p \\\\ \underbrace{p^3+7}_{=1007_p}+\underbrace{3p^2+6}_{=306_p}+ \underbrace{p^2+p+3}_{=113_p} + \underbrace{p^2+2p+5}_{=125_p} +\underbrace{6}_{=6_p} &=& \underbrace{p^2+4p+2}_{=142_p}+ \underbrace{2p^2+7p+1}_{=271_p} + \underbrace{3p^2+6p}_{=360_p} \\ p^3+ 3p^2+ p^2+ p^2 +p +2p+7 +6 +3 +5 + 6 &=& p^2+ 2p^2+ 3p^2 +4p+ 7p+6p +2 + 1 \\ p^3+ 5p^2+3p +27 &=& 6p^2 +17p +3 \\ p^3+ 5p^2-6p^2+3p-17p +27-3 &=& 0 \\ p^3-p^2-14p +24 &=& 0 \\ (p-2)(p^2+p-12) &=& 0 \\ (p-2)(p-3)(p+4) &=& 0 \\ \hline \end{array} }\)
The only primes that will work are \( p=2 \) or \(p=3\)
