Determine the respective smallest and largest 3-digit palindromes in base 2017 each of which is divisible by 7.
*** Leading zeros in any of the palindromes is not allowed.
2016 is divisible by 7, so 2107=1mod7
Let the digits in base 2017 be (a,b,a)
the smallest would be (1,b,1) with b as small as possible
2017^2 + 1 = 2mod7 so if b=5 we'd add 5mod7 and get the number
(1,5,1)
checking in decimal this equals 4078375 = 7*582625
the greatest would be (2016,b,2016) with b as large as possible
2016*2017^2 + 2016 = 0 mod 7
any value of b would preserve this so b=2016 would be the largest:
(2016,2016,2016)
checking in decimal this equals 8205738912 = 7*1172248416
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Posted by Jer
on 2016-04-11 09:29:48 |
Analytic solution
