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Finding the Unit Digits of Powers of Integers |
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Here are the values of the unit digits for the first five powers of the digits.
The Unit Digits of the First Five Powers |
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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.
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.
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.
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