To find the last digit of a number raised to a power, you only need to look at the last digit of the base number and the last two digits of the exponent. Because the last digits of powers repeat in a cycle, you can use modular arithmetic to determine where in that cycle the specific power will land.
The process in three steps
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Identify the last digit of the base. Discard all other digits of the base number. For example, for 123456123 to the 456th power
123456
, the base is 123 and the last digit is 3.
-
Examine the exponent and its pattern. Find the remainder of the exponent when divided by 4. This reveals where the power falls in its cycle. Use the last two digits of the exponent to make this division simpler.
- Special case: If the remainder is 0, the position is the same as the 4th power in the cycle.
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Find the final last digit. Raise the last digit of the base (from step 1) to the power of the remainder you found (from step 2). The last digit of this result is your answer.
Last digit patterns of powers
The core of this method relies on the fact that the last digits of powers repeat in short cycles. Here is a breakdown of the patterns for every possible last digit of the base:
Last digits of powers follow repeating patterns. For bases ending in 0, 1, 5, or 6, the last digit is always the same as the base's last digit. For bases ending in 4 or 9, the pattern repeats every two powers. For bases ending in 2, 3, 7, or 8, the pattern repeats every four powers. To find the position in the cycle for these bases, you find the remainder of the exponent when divided by 4.
Example walkthrough: 117135117 to the 135th power
117135
Let's find the last digit of 117135117 to the 135th power
117135
.
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The last digit of the base (117) is 7.
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The exponent is 135. We find the remainder of 135 divided by 4: 135÷4=33135 divided by 4 equals 33
135÷4=33
with a remainder of 3.
-
We raise the base's last digit (7) to the power of the remainder (3): 73=3437 cubed equals 343
73=343
. The last digit is 3.
So, the last digit of 117135117 to the 135th power
117135
is 3.
Example with a remainder of 0: 19288192 to the 88th power
19288
Here's how to find the last digit of 19288192 to the 88th power
19288
.
-
The last digit of the base (192) is 2.
-
The exponent is 88. We find the remainder of 88 divided by 4: 88÷4=2288 divided by 4 equals 22
88÷4=22
with a remainder of 0. Since the remainder is 0, we use 4 as the power.
-
We raise the base's last digit (2) to the power of 4: 24=162 to the fourth power equals 16
24=16
. The last digit is 6.
The last digit of 19288192 to the 88th power
19288
is 6.
Why does this work? The role of modular arithmetic
This method works due to modular arithmetic. Finding the last digit is equivalent to finding the remainder when a number is divided by 10. A key property is that the last digit of a product is the same as the last digit of the product of the last digits of the original numbers. This extends to powers. The repeating patterns arise from this behavior. By using the remainder when the exponent is divided by the cycle length (which is often 4), we can quickly determine the last digit without calculating the full power.