For matrix with full column rank, QR algorithm is among the best approach to solve wider class of least squares problem (LS). Using the communication optimal variant of TSQR, we study the scalability of the least squares solver with multiple right hand sides. The communication for TSQR based LS solver for multiple right hand sides is still optimal in the sense that no additional messages are necessary compared to TSQR. However, LS has additional communication volume, and flops compared to that for TSQR. The scalability of the proposed method is studied up to few thousand cores using global address space programming framework (GPI) and pthreads.