Consider the following example expression from modular arithmetic:
x² ≡ y (mod n)
It is read aloud as
X Quadrat ist äquivalent zu Y modulo N.
Here, modulo denotes the Modulo operation, and n is referred to as der Modulus, the plural of which is die Moduli. Actually, modulo is the ablative case of the Latin word modulus. Gauß himself used Modulus/ Moduli, not only in his Latin work Disquisitiones Arithmeticae, but also in German texts (see, for example, various passages in Carl Friedrich Gauß. Werke. Band II). Nowadays, some people refer to n as der Modul (with stress on the first syllable), the plural of which is die Moduln. For example, in his famous book „Das ist o. B. d. A. trivial!“ Beutelspacher (2009, 9th edt., p. 81) writes:
„Man nennt die Zahl n den Modul. Früher hieß dies (auf Lateinisch) der modulus; daher kommt der traditionelle Plural die Moduli; heute kann man aber
auch Moduln sagen.“
Personally, I stick with Gauß and use Modulus/ Moduli, when I mean the concept in modular arithmetic. In the realm of linear algebra, I use Modul/ Moduln to denote Abelian groups that are acted upon by elements of a commutative ring.
As a last point, there is the term das Modul (with stress on the second syllable), the plural of which is die Module. Outside of mathematics, it is used to refer to any kind of modular unit.