What’s the correct usage of “modulo”?
Mathematics would say: "10 is congruent to 1 modulo 3". The modified usage "10 modulo 3" with "modulo" as an operation (like "10 plus 3") is from computer programming.
The verb in your sentence is equals. The entire sentence can be parsed either as 10 modulo 3 being a noun phrase with modulo 3 operating as an adjective, or as modulo 3 equals being a verb phrase with modulo 3 operating as an adverb. Whichever analysis is preferred, the usage is perfectly standard, if slightly imprecise. The reference you cite is correct to prefer is congruent to over equals.