The procedure:
Take any number (Say x).
Form a set of numbers by adding 10 (Your set={x,x+10,x+20,x+30,...}).
Raise every member of the obtained set to its sixth power.
Observe the last two digits (taken from right to left) of the obtained sixth powers.
The last two digits of the sixth powers repeat after every five members of the set.
(Note the numbers highlighted in yellow repeating in the same column)
The Proof:
Any 6 terms of the series will be of the format x,x+10,x+20,x+30,x+40,x+50.
We need to prove that the pattern of the sixth powers repeats after every 5 terms.
Therefore, it suffices to prove that the digits at the unit place and tens place of sixth power of a number are equal when the number after five terms is raised to the sixth power.
Therefore, we need to prove that the last two digits of the sixth power of x and the sixth power of x+50 are equal. (Since the numbers are taken at a difference of 10 and there are 5 such terms.)
Similarly, last two digits of sixth powers of x+10 and x+60 will be equal.
Similarly, last two digits of sixth powers of x+20 and x+70 will be equal.
And so on ...
Thus, the pattern continues to repeat after every five terms.
Derived Corollary:
The last two digits of a number's sixth power and another number's sixth power are the same when another number is formed by adding a multiple of 50 to the first number.
Important Observation:
If any number has 5 at its unit place, its sixth power ends with 2 and 5 at ten's and unit's place respectively. (This, too can be proved easily).
A similar pattern is observed in the squares, cubes, fourth powers and fifth powers of numbers.
Q.E.D
© Rishabh Bidya
Take any number (Say x).
Form a set of numbers by adding 10 (Your set={x,x+10,x+20,x+30,...}).
Raise every member of the obtained set to its sixth power.
Observe the last two digits (taken from right to left) of the obtained sixth powers.
The last two digits of the sixth powers repeat after every five members of the set.
The Proof:
Any 6 terms of the series will be of the format x,x+10,x+20,x+30,x+40,x+50.
We need to prove that the pattern of the sixth powers repeats after every 5 terms.
Therefore, it suffices to prove that the digits at the unit place and tens place of sixth power of a number are equal when the number after five terms is raised to the sixth power.
Therefore, we need to prove that the last two digits of the sixth power of x and the sixth power of x+50 are equal. (Since the numbers are taken at a difference of 10 and there are 5 such terms.)
Similarly, last two digits of sixth powers of x+10 and x+60 will be equal.
Similarly, last two digits of sixth powers of x+20 and x+70 will be equal.
And so on ...
Thus, the pattern continues to repeat after every five terms.
Derived Corollary:
The last two digits of a number's sixth power and another number's sixth power are the same when another number is formed by adding a multiple of 50 to the first number.
Important Observation:
If any number has 5 at its unit place, its sixth power ends with 2 and 5 at ten's and unit's place respectively. (This, too can be proved easily).
A similar pattern is observed in the squares, cubes, fourth powers and fifth powers of numbers.
Q.E.D
© Rishabh Bidya
