Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 230

must be 28

Show that if

is odd,

then the last 2 digits of

must be 28.

Last edited by tony123 (2008-09-20 08:44:12)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: must be 28

First show that 

is divisible by 20 for all positive integers k. This is straightforwardly done by induction, so I’ll omit the proof and let you do it yourself.

So

i.e.

i.e.

Now to the original problem. If you write

, the expression becomes

When

, this is 28.

Assume that

for a positive integer k. Then

As

, the above is congruent to

modulo 100, and the result is proved by induction.