The procedure:
Take any number (Say x).
Form a set of numbers by adding 20 (Your set={x,x+20,x+40,x+60,...}).
Raise every member of the obtained set to its fifth power.
Observe the last two digits (taken from right to left) of the obtained fifth powers.
The last two digits of the fifth powers of every member of the set are the same.
The Proof:
Any term of the series will be of the format x+20k(where k must be a non negative integer).
We need to prove that the digits at the unit's and tens's place of the fifth powers of x and x+20k are same.
Example:
We choose 3 as our x.
The set of numbers obtained is {3,23,43,64,...}.
We raise the terms of this set to their fifth powers {243,6436343,147008443,992436543,...}.
We observe the last two digits of the obtained fifth powers.
The number at unit's place is 3 whereas the number at the tens' place is 4.
Derived Corollary:
The last two digits of an odd number's fifth power and another number's fifth power are the same when another number is formed by adding a multiple of 20 to the first number.
Q.E.D
© Rishabh Bidya
© Rishabh Bidya
