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1001 is the lowest four-digit palindrome, which is 0 modulus 7. Adding one to the first and last digits adds 0 modulus 7. Adding one to the second and third digits adds 5 modulus 7, since 110 is also 5 modulus 7. So only 770 plus multiples of 1001 are divisible by 7.

Out of 90 possibilities, there are only 9 that are divisible by 7. So the probability is 9/90, or 1/10.

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Q: If a number is chosen at random from all four digit palindromes what is the probability that it is divisible by seven?
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