Let n be an even integer. Defines an n-digit tautonym as an n-digit integer in which the first n/2 digits are in the same order as the last n/2 digits. For example, 4,721,248,712 is a 10-digit tautonym. What the least common and greatest common prime divisor for the set of 10-digit tautonym and prove your answer?

The greatest common divisor of all 10-digit tautonims is 100001 and the greatest prime divisor is 9091.

Expert Answers

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First, note that 4,721,248,712 is not a 10-digit tautonym. Its first half is 47212 and the last is 48712, but they must be the same. For example, 4721247212 is a tautonim.

Now determine what arithmetic operation is used, if any, to produce a tautonim from its last part, i.e. from a 5-digit number, say, `a b c d e :` actually, the number `a b c d e a b c d e` is equal to

`a b c d e 0 0 0 0 0 + a b c d e = a b c d e + a b c d e * 100000 = a b c d e * 100001 .`

Now we see that all 10-digit tautonims have a common divisor `100001 .` Prove that it is the greatest common divisor of them. Indeed, consider any two different 5-digit primes p and q. The tautonims made from them are equal to `1 0 0 0 0 1 p` and `100001 q,` which clearly have no larger common divisors than `100001 .`

The number `1 0 0 0 0 1` is clearly divisible by `11` and `100001 / 11 = 9091 ,` which is a prime number.

This way, the greatest common divisor is `100001` and the greatest common prime divisor is 9091.

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