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What are the last 6 digits of this number: $5^{131313}$ ?
 Jun 21, 2018
 #1
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5^131313 =Here are the first 100+ digits and the last 100+ digits:

 

7049 8418681817 1834309048 3362468894 0890283325 3078192378 2991630210 0084853135 0584466688 5191910408 0441239137..........................................................................................
 5734346081 1680875733 7217652363 8489725205 4558965700 2641348330 8060680754 0139137831 8281387849 8695903118 6870561214 1638994216 9189453125

It turns out that 5^(13+4+4^n), where n =>2, will give the same last 6 digits:453125.

 Jun 21, 2018
edited by Guest  Jun 21, 2018
 #2
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Here is the general rule that gives the same last 6 digits of 453125.....

 

5^(16n + 1), for n =1, 2, 3........etc. 131313 is a multiple of 16n + 1, where n =8,207.

 Jun 22, 2018

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