Question:

Whenever I run: sudo apt-get update, I get the following messages:

~ sudo apt-get update
.................................
Fetched 208 kB in 36s (5,644 B/s)
W: Failed to fetch gzip:/var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_saucy_universe_binary-amd64_Packages  Hash Sum mismatch

W: Failed to fetch gzip:/var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_saucy_main_binary-i386_Packages  Hash Sum mismatch

W: Failed to fetch gzip:/var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_saucy_universe_binary-i386_Packages  Hash Sum mismatch

E: Some index files failed to download. They have been ignored, or old ones used instead.

All you simply need to do is to take note of the location, in this case /var/lib/apt/lists/partial/, The open up your terminal and remove all files residing in there using the commands below:

sudo rm -r /var/lib/apt/lists/partial/*  
sudo rm -r /var/lib/apt/lists/*

Next time you run sudo apt-get update it should work as required:

~ sudo apt-get update
...............................................................
Ign http://archive.ubuntu.com saucy/restricted Translation-en_US
Ign http://archive.ubuntu.com saucy/universe Translation-en_US
Ign http://archive.ubuntu.com saucy-updates/main Translation-en_US
Ign http://archive.ubuntu.com saucy-updates/multiverse Translation-en_US
Ign http://archive.ubuntu.com saucy-updates/restricted Translation-en_US
Ign http://archive.ubuntu.com saucy-updates/universe Translation-en_US
Fetched 20.5 MB in 10min 11s (33.6 kB/s)
Reading package lists... Done

Enjoy!