Sunday, May 23, 2010

GR9768.61: Greatest Integer, Prime

61.  What is the greatest integer that divides $p^4 -1$ for every prime number p greater than 5?
(A) $12$
(B) $30$
(C) $48$
(D) $120$
(E) $240$

Solution:
$p^4 -1 = (p-1)(p^3 + p^2 + p + 1)$.
And we see that $(p-1)$ and $(p^3 + p^2 + p + 1)$ are the divisors of $p^4 -1$.
Guess and check:
$p = 13$ works so A is a divisor.
$p = 31$ works so B is a divisor.
$\vdots$
$p = 241$ works so E is a divisor.
And $p-1 = 240$ is the greatest integer provided.