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Number Theory Level 1

We know that 13 and 31 are both prime. If we combine them together to get 1331 or 3113, then neither is prime.

Now, consider a b ‾ \overline{ab} and b a ‾ \overline{ba} , where a a and b b are both positive single digits. Can we find some values of a a and b b such that a b ‾ , b a ‾ , a b b a ‾ , b a a b ‾ \overline{ab},\, \overline{ba},\, \overline{abba},\, \overline{baab} are all primes?

Yes, it is possible No, it's not possible

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Michael Huang by Michael Huang , 29, USA

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2 solutions

Kenny O.
Oct 6, 2017

a b b a = a 00 a + b b 0 = a ∗ 11 ∗ 91 + b ∗ 11 ∗ 10 = 11 ( 91 ∗ a + b ∗ 10 ) abba= a00a+bb0= a*11*91+b*11*10=11(91*a+b*10) . abba is not prime as it is away divisible by 11. Using the same logic for baab, there are no such prime numbers.

10 Helpful 0 Interesting 1 Brilliant 0 Confused
Luis Salazar
Oct 24, 2017

For every of this 4 digit palindrome, like in the form '1331', it will always be divisible by 11.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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