Inspired by the Noether-Lefschetz theorem, it has been conjectured that the degree of any curve on a (very) general complete intersection variety is divisible by the degree of the variety itself. In this talk, we present a weaker version of this conjecture. Using this result, we confirm a conjecture by Bastianelli--De Poi--Ein--Lazarsfeld--Ullery regarding measures of irrationality. This is joint work with Nathan Chen and Benjamin Church.