Dice and Division

Player \( A \) and player \( B \), are playing a game. They each have an unbiased 10-sided dice with faces labelled with the numbers \( 1,2,3,...,10 \). To play, \( A \) and \( B \) simultaneously roll their dice. The number that \( A \) rolls is denoted as \( m \) and the number that \( B \) rolls is denoted as \( n \). If \( m = n \) then they roll the dice again (ie. it is like the last roll didn't happen). If \( |m-n| \) divides \( m+n \) then \( B \) wins, else \( A \) wins. The probability that \( B \) wins can be expressed as \( \dfrac{p}{q} \) where \( p \) and \( q \) are coprime positive integers. What is \( p+q \)?

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