Any one of these answers would be correct. There really are two Boulder Rivers in Montana. Not only do both rivers have the same title, but they also both discharge nearly the same amount of water. The daily-mean water-year count of the Boulder River at Big Timber flows at 23,226 cf/s annually, and the Boulder River at Boulder has a daily-mean water-year count of 25,440 cf/s. This though, is largely where the similarities end.
There is a reason the Boulder River of Southern Montana is the more well known of the two. It begins at nearly 9,000 ft. in the high country of the Absaroka-Beartooth Wilderness, flowing north over 50 miles. The river actually forms the dividing boundary between these two great ranges. Not only does it offer excellent fly fishing, it contains Class V rapids and one of the most beautiful waterfalls Montana has to offer, Natural Bridge Falls. A marginally insane kayaker has even gone over this impressive cataract. The valley it flows north through truly is breathtaking, with some of Montana's mightiest peaks creating the walls that surround it. Parts of the movie, "A River Runs Through It" were filmed here. Along its banks, dilapidated cabins show evidence of the first white occupants of the valley. Eventually, the river leaves the canyon and flows through the agricultural high plains that border the northern side of the mountains. Near the community of Mcleod, the West and East Forks of the river pour into the Boulder's mainstem. Around 15 miles north of Mcleod, the river's mouth can be found near Big Timber, on the southern side of the Yellowstone River. In 1806, Captain Clark named this location, "Rivers Across".
If from the mouth of Southern Montana's Boulder River, we head 150 miles northwest, to the Continental Divide along the Mountains of the Boulder Batholith. We will find ourselves near the headwaters of the other Boulder River. Originally named Field's Creek, by Lewis and Clark, it begins by flowing north off of the divide separating Silver Bow and Jefferson Counties. Curving to the east, it absorbs most of its water from the north, provided by snow melt and springs along the southern face of the proposed Electric Peak Wilderness area. Between Bernice and Boulder, Montana, it flows straight and fast along Interstate 15. Along this stretch the remnants of mining are frequent. The communities of Basin and Comet are both residual districts of a bygone mining era. Also present along the banks of this Boulder River are numerous Radon Health Mines. Many people go sit in these mines to breathe in small amounts of what is typically a cancer causing, radioactive gas, to cure what ails them. To each their own? At the town of Boulder, the river curves south and leaves I-15 and the mountains. It then meanders south through the beautiful Boulder Valley. In late spring the Boulder River rages, with plenty of snow to feed it. But in late summer the river below the town of Boulder is chronically dewatered, and often runs dry only to be replenished further near its mouth. This is a byproduct of irrigation needs, and over-allocation, by way of Montana's Prior Appropriations system. For nearly its entire length though, when the water allows, this Boulder River also offers some of Montana's most overlooked and under appreciated fly fishing. Around 30 miles south of Boulder, the river's mouth can be found near Cardwell, Montana. Here it empties into the north side of the Jefferson River.
On one hand, it's a shame that these two rivers share a name. They both have something unique to offer, but much of it gets lost in the confusion that follows from duplicate toponyms. On the other hand though, the names are fitting. Both rivers are filthy with their namesake, and independently the titles fit. It would be a good thing though, if more Montanans fell into the, "Which one?" category, instead of short-changing a river or themselves. Because until you've taken in both Boulder Rivers, you really are missing out.