What is the units digit in the number (123457)^655?

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The units digit in `(123457)^655` has to be determined.

`(123457)^655`

=> `(123457)^654*123457`

=> `(123457)^(2*327)*123457`

=> `(123457^2)^327*123457`

The units digit of `123457^2` is 9 as `7^2 = 49`

`(123457^2)^327*123457`

=> `(XXX9)^327*123457`

=> `(XXX9)^326*XXX9*123457`

=> `((XXX9)^2)^163*XXX9*123457`

The units digit of `(XXX9)^2` is 1 as `9^2 = 81` . The units digit of the power of any number ending with 1 is 1 as 1 raised to any power is 1.

=> `XX1*XXX9*123457`

=> `XXXX9*123457`

The units digit of the product arrived at above is 3 as 9*7 = 63

**The units digit of `(123457)^655` is 3**

What is the units digit for `(123457)^655` :

`7^0=1,7^1=7,7^2=49,7^3=343,7^4=2401,7^5=16807` ...

Notice the units digit: it runs in a cycle 1,7,9,3,1,7,9,3,... Every 4th power of a number ending in 7 has a units digit of 1. (So `7^4` ends in 1, `7^44` ends in 1, etc...)

In particular, `(123457)^652` ends in 1. So extending the pattern if the exponent is 653 it will end in 7, 654 ends in 9, and 655 ends in 3.

**The units digit of `(123457)^(655)` is a 3.**

This idea works for any number. The units digits will cycle. Another example would be `123453^655` : powers of 3 cycle 1,3,9,7,1,3,9,7,... so this number ends in 7.

Numbers ending in 2 cycle 2,4,8,6,2,4,8,6,...

Numbers ending in 4 cycle 4,6,4,6,...

Numbers ending in 8 cycle 8,4,2,6,8,...

Numbers ending in 0,1,5,6 are easy since they end in 0,1,5,6 respectively for any power.

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