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This work considers a distributed source coding (DSC) problem where L encoders observe noisy linear combinations of K correlated remote Gaussian sources with K ≤ L, and separately transmit the compressed observations to the decoder to reconstruct the remote sources subject to a sum-distortion constraint. This DSC problem can be viewed as a generalization of the quadratic Gaussian CEO problem with only one remote source; it is also an extension of Oohama's latest work with correlated Gaussian remote sources, where the number of remote sources equals the number of observations. For our new DSC problem, we first provide general inner and outer rate regions, followed by a tighter outer rate region when the transform matrix between the sources and the encoders is semi-orthogonal. We then give a sufficient condition - in the form of capping the target distortion under certain threshold - for our inner and outer rate regions to match. When K = 1, both our inner and outer regions specialize to known result for the Gaussian CEO problem; with K = L and identity transform, our inner and outer regions degenerate to those provided by Oohama, however, our sufficient condition is more relaxed in the sense of allowing more matching cases.