One arcminute is exactly 60 arcseconds. The arcsecond is the working unit of positional astronomy, and it is small enough that the entire scale of the universe outside the solar system is built on measuring it.
About the units
An arcsecond is a sixtieth of an arcminute and therefore a 3,600th of a degree, written with a double prime: 5″. It is a very small angle — about the size of a coin seen from four kilometres away. Its importance comes from parallax: a star one parsec distant shows a parallax of exactly one arcsecond as the Earth moves across its orbit, which is the definition of the parsec and the reason the unit exists. No star is that close; the nearest, Proxima Centauri, has a parallax of 0.77 arcseconds.
The exact factor
The factor is exactly 60, and a degree is 3,600 arcseconds. The scale is what makes the unit interesting. Atmospheric turbulence limits ground-based telescope resolution to about one arcsecond at a good site, which is why space telescopes were transformative — Hubble resolves 0.05 arcseconds. The Gaia spacecraft measured stellar positions to about 25 microarcseconds, which is roughly the width of a human hair seen from a thousand kilometres. Milliarcseconds and microarcseconds are standard units in modern astrometry.
Where you meet this conversion
Astronomy and geodesy. Star catalogues give positions in right ascension and declination to fractions of an arcsecond. Telescope pointing accuracy, seeing conditions and the apparent separation of double stars are all quoted in arcseconds. In surveying, high-precision theodolites read to one arcsecond, and GPS-derived coordinates are given in degrees, minutes and seconds where an arcsecond of latitude is about 31 metres. Optical engineering specifies wedge and alignment tolerances in arcseconds.