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The interstellar oxygen isotopic ratio of 18O/17O can reflect the relative amount of the secular enrichment by ejecta from high-mass versus intermediate-mass stars. Previous observations found a Galactic gradient of 18O/17O, i.e., low ratios in the Galactic center and large values in the Galactic disk, which supports the insideout formation scenario of our Galaxy. However, the observed objects are not many and, in particular, not so many at large galactocentric distances. Thus we started a systematic study on Galactic interstellar 18O/17O, through observations of C18O and C17O multi-transition lines toward a large sample of 286 sources (at least one order of magnitude larger than previous ones), from the Galactic center region to the far outer Galaxy (~22 kpc). Here we present our observations of J=1-0 lines of C18O and C17O, with the ARO12m and the IRAM 30m telescope. We detected successfully both C18O and C17O 1-0 lines for 34 sources among our IRAM30m sample of 50 targets and for 166 sources among our ARO12m sample of 260 targets. The C18O optical depth effect on our ratio results, evaluated by fitting results of C17O spectra with hyperfine components and our RADEX non-LTE model calculation for the strongest source, was found to be insignificant. Beam dilution does not seem to be a problem either, which was supported by the fact of no systematic variation between the isotopic ratio and the heliocentric distance, and consistent measured ratios from two telescopes for most of those detected sources. Our results, though there are still very few detections made for sources in the outer Galaxy, confirm the apparent 18O/17O gradient of 18O/17O = (0.10+-0.03)R_GC+(2.95+-0.30), with a Pearson's rank correlation coefficient R = 0.69. This is supported by the newest Galactic chemical evolution model including the impact of massive stellar rotators and novae.