The Chuncheon landslide, which witnesses said sounded like a massive explosion, killed a married couple and shopkeeper as well as the students, who were in the area for to do voluntary work. Hundreds had to leave their homes and 24 others were injured.
Nearly 700 firefighters, police and other rescue workers have been rushed to the area.
In southern Seoul 16 people died when mud crashed into homes at the foot of a mountain.
Three others after a stream just south of the city flooded.
Ten people are reported missing across the country, according to the National Emergency Management Agency.
Muddy water filled the streets of Seoul Wednesday, forcing some residents to perch on the roofs of their partially submerged cars, flooding metro stations and causing sewers to overflow. Three people were missing after they were swept away in flash floods.
Heavy rain that started Tuesday left about 620 people homeless and flooded 720 houses and about 100 vehicles throughout South Korea, the emergency management agency said.
About 440 millimetres of rain fell on Seoul and more than 340 millimetres on Chuncheon on Tuesday and Wednesday, about 15 times more than the average two-day rainfall at this time of year, according to the state-run Korea Meteorological Administration.
Weather officials said another 254 millimetres could fall in northern South Korea, including Seoul, up until Friday.