LogMeIn Inc. made a name for itself a decade ago, by offering its computer remote control software for free. But that’s about to end. The Boston company says it’s phasing out the free version of its service, that lets people operate their home or office computers remotely when they’re on the road.

LogMeIn spokesman Craig Vercolen said the free service served a purpose in the company’s early days. “It got us on the map, gave us a huge footprint,” he said. It worked; today LogMeIn is a publicly-traded company with a market capitalization of about $835 million.

But the company is now changing its strategy in response to a transformation in the way people use their computers. Many people use LogMeIn mainly to get access to files stored on remote computers. But these days, millions store their files on cloud-based services like Dropbox and SkyDrive, enabling them to get their files on any Internet-connected computer, tablet or smartphone. This means fewer people need to log onto their personal machines when they’re traveling. “The need for remote access has completely changed,” said Vercolen. “The market has matured.”

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However, LogMeIn still makes sense for people who need to run software that’s installed on remote machines. For these users, the product will remain available for $99, which entitles the user to install the program on two personal computers and two mobile devices. LogMeIn is offering discounts of up to 50 percent to customers who quickly sign up for paid service. When users of the free service log in, they’ll see an on-screen message announcing the new pricing policy.

Vercolen would not say how many people use the free version of LogMeIn, but said the company has about 20 million users for its various products. These include a paid version of LogMeIn used by technology companies to do remote maintenance on customers’ computers, and Cubby, a cloud-based file storage service similar to Dropbox.