Re: a problem in elementary number theory

In article
<1193777979.713787.161730@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
,
"dmitry.sustretov@xxxxxxxxx"
<dmitry.sustretov@xxxxxxxxx> wrote:

Hello,

I am stuck solving this problem from GRE Math training booklet:

Find the maximal integer x such that x divides p^4-1 for all prime
numbers p > 5.

[they actually have a list to choose from: 12, 30, 48, 120, 240]

Do you have any ideas?

p^4 - 1 = (p^2 + 1)(p^2 - 1)
Set p = 2k + 1.
p^2 - 1 = 4.nn + 4.n + 1 - 1 = 4.n(n +1) == 0 (mod 8)
p^2 + 1 == 2 (mod 8)

so that eliminates answers 12, 30, 120.
Now guess.

Or if you have more time p^2 = +/- 1 (mod 5)
therefore the answer is 240.

--
Michael Press
.

Relevant Pages
Loading