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We consider the distributed source coding system for L correlated memoryless Gaussian remote sources specified with L correlated Gaussian random variables Xi, i=1,2,...,L. We deal with the case where each of L distributed encoders cannot directly observe the source Xi but its noisy version Yi=Xi+Ni. Here, Ni, i=1,2,..., L are independent additive L Gaussian noises also independent of Xi , i=1,2,..., L. On this coding system, the determination problem of the rate distortion region remains open in general. In this paper, we derive explicit outer and inner bounds of the rate distortion region. We further find an explicit sufficient condition for those two bounds to match. We also study the sum rate part of the rate distortion region when the correlation has some symmetrical property. We derive an explicit upper bound of the sum rate part from the inner bounds of the rate distortion region. On a lower bound, we derive a new explicit bound by making full use of the symmetrical property of the correlation. We further derive an explicit necessary and sufficient condition for the lower bound to coincide with the upper bound.