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Find the remainder when $100^{100}$ is divided by $18$.

 Jul 24, 2024
 #1
avatar+16 
+1

100100=(102)100=10200.
Now we can check the remainder of the powers of 10 being divided by 18.

10210mod(18)102(185)=10

10310mod(18)103(1855)=10

10410mod(18)104(18555)=10

10510mod(18)105(185555)=10

10610mod(18)106(1855555)=10

...

Therefore we can conclude that the remainder of 100100 divided by 18 is 10

 Jul 24, 2024
 #2
avatar+33652 
+3

You could also look at it this way:

 

 

You can see that this will always end with 10/18, i.e. there will always be a remainder of 10.

 Jul 24, 2024

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