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You can do this on any browser of your choosing. If you're using the Slack app for Android, iPhone, or Windows phone, tap the app to open Slack. This will redirect you to a team name entry field. On mobile, this should say "Sign in to an existing team". This should take you to your team's Slack conversation. On mobile, tap "Go" to continue. You'll also have to enter your Slack-associated email address and its corresponding password. Each channel contains a conversation, whether between two individuals or an entire group. Channels are listed on the left side of the screen. On mobile, swipe right to open the channel menu, then tap the channel you want to mute. This menu is located to the left of the search bar at the top of the screen. On mobile, tap the "#[channel name]" in the top left of your screen to open the settings menu. This will prevent you from receiving notifications from that particular channel; however, you will still be able to access the channel at any time. On mobile, tap the "Notifications" option, then tap "Mute Channel" at the bottom of the menu. You can unmute the channel by tapping the "Mute Channel" option again. Going back to the Channel Settings menu and clicking "unmute #[channel name]" will restore notifications. You can also click the bell with a slash through it next to the channel name whilst in the channel itself to unmute the channel.

Summary:
Open Slack's website. Click "Sign in" at the bottom of the page. Enter your team's name, then click "Continue". Click on the channel you wish to mute. Click on the gear symbol to open the Channel Settings menu. Click the "mute #[channel name]" option to mute the channel. Unmute the channel when you're ready.