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Finding the Unit Digits of Powers of Integers

Here are the values of the
unit digits for the first five powers of the digits.

.

The Unit Digits of the First Five Powers

Powers

1

2

3

4

5

1

0

0

0

0

0

1

1

1

1

1

1

1

2

4

8

6

2

1

3

9

7

1

3

1

4

6

4

6

4

1

5

5

5

5

5

1

6

6

6

6

6

1

7

9

3

1

7

1

8

4

2

6

8

1

9

1

9

1

9

Since the unit digits for the fifth power are the same as those of the first power the cycle starts over again. The cycle has a period
of four. Thus the unit digits for the ninth power are the same as those of the first and fifth. Likewise this is true for the
powers 13, 17, 21, 25 and so forth.

Note that the unit digit of any power of a number ending in 5 is 5 and of any number ending in 6 is 6.
The same is true for
the digits 0 and 1.

In the table there are three cases for which the unit digits of all powers is the same as the base; i.e., 0, 1, 5 and 6. There
are two cases in the cycle in digits is two digits long; i.e., 4 and 9. The rest have cycles of period 4.

Finding the Unit Digit of a Power of an Integer

To get the unit digit of a number m raised to the K-th power, first step is to find the remainder for K when divided
by 4. If that remainder is 0 then 4 is used instead of 0. Say this modified remainder is h. Then if n is the unit digit of m the unit digit of m^{K} is read from
the above table for the n-th row and h-th column. For example, suppose m is 27 and K is 75. Then h is 3 and
n is 7. The value found in the row for 7 and the column for 3 is 3. Thus the unit digit of 27^{75}
is 3.

Special provision must be made for the zeroeth powers of numbers. They are all unity.

Finding the Unit Digit of an Exact Root of of an Integer Raised to a Power

For some values of h it is possible to determine uniquely the unit digit of the integer which was raised
to the power. Those are values of 1 and 3. For the case of h=1 the situation is simple; the unit digit of the
base and power are the same. For the case of h=3 the determination of the unit digit is not so simple
but is easily done.

For h equal to 2 and 4 it is not possible to uniquely determine
the unit digit. For example, for h equal to 2 and the unit digit of the power is 9. Then the unit digit of the
base could be 3 or 7.