It's not entirely a coincidence but also not a general result that one should use the modular inverse as multiplier.
171 * 3 = 2^9 + 1, which is not surprising as we know that 171 * 3 = 1 (mod 2^8). So rearranged we have 171 / 2^9 = 1/3 + 1/(2^9*3) which shows it's a close approximation of 1/3.
171 * 3 = 2^9 + 1, which is not surprising as we know that 171 * 3 = 1 (mod 2^8). So rearranged we have 171 / 2^9 = 1/3 + 1/(2^9*3) which shows it's a close approximation of 1/3.