Maine's State Route 113 runs north-south near the state line north of Fryeburg
Maine. In the process, it enters into New Hampshire twice, at East Conway
for about a mile, and near North Chatham for about 5 miles. It bears Maine
style signage in New Hampshire.New Hampshire 113B starts in Maine at Stow, runs west to Chatham, New Hampshire, then north, rejoining Maine 113 right where 113 enters New Hampshire. It bears New Hampshire style signage, thus is a New Hampshire spur route of a Maine state route.
If you aren't confused yet, New Hampshire Route 113 is also nearby, it meets US 302 in Conway, and is cosigned with 302 to the state line. At this point it is about a mile away from where Maine's 113 meets US 302, and about a mile away from where Maine's 113 enters New Hampshire in East Conway. You can't consider Maine's 113 as a continuation of NH 113, since Maine 113 also runs south of Fryeburg.