Tower of Nine
by luimichael, Apr 6, 2017, 12:32 PM
Find the last two digits of
and
.
Solution
Remark
In view of the above proof, we see that the last two digits of the tower of 9 (to any order > 2) will be 89


Solution
We first show that
---(*)
To see this , consider
by binomial expansion.
. by binomial expansion
We may let
for some positive integer k.
Then
. ( we have applied (*) here.)
But
Hence, we get
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We shall also show that
.
By the above result, we may let
for some positive integer s.
Then
by (*) and the above results.

To see this , consider


We may let

Then

But

Hence, we get

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We shall also show that

By the above result, we may let

Then

by (*) and the above results.
Remark
In view of the above proof, we see that the last two digits of the tower of 9 (to any order > 2) will be 89