One degree is exactly 60 arcminutes. The sexagesimal subdivision is the same one used for time, from the same Babylonian source — and in navigation the two are directly linked, because one arcminute of latitude is one nautical mile.
About the units
An arcminute is a sixtieth of a degree, written with a prime symbol: 30′ is half a degree. It subdivides further into 60 arcseconds. The system is identical in structure to hours, minutes and seconds of time, and both descend from Babylonian base-60 counting. The link to distance is what makes the arcminute genuinely useful rather than merely traditional: because a nautical mile was defined as one minute of latitude, a navigator reading a chart converts angle to distance by inspection, with no arithmetic at all.
The exact factor
The factor is exactly 60. Some reference points that give a feel for the scale: the Moon and the Sun both appear about 30 arcminutes across from Earth, which is half a degree — the coincidence that makes total solar eclipses possible. Normal human visual acuity resolves about one arcminute, which is what the 20/20 line on an eye chart tests. A degree of latitude is 60 nautical miles or about 111 kilometres, so one arcminute is 1.852 kilometres anywhere on Earth.
Where you meet this conversion
Navigation and astronomy. Latitude and longitude are written in degrees and decimal minutes in marine practice — 59° 19.8′ N — because the minutes convert directly to nautical miles. Aviation charts use the same convention. In astronomy, the apparent sizes of the Moon, planets and deep-sky objects are quoted in arcminutes. Optometry measures visual acuity and astigmatism axes in the same units, and rifle sights are adjusted in fractions of a minute of angle, where one MOA is about one inch at 100 yards.