What is the largest four-digit negative integer congruent to $1 / pmod{23}?$

-1011

Step-by-step explanation:

An integer that is congruent to 1 mod{23} is of the form 23n+1.

Therefore, we form the inequality 23n+1<-999, and find the largest possible integer n. We get

23n+1<-999

23n<-1000

n < 1000/23, or n < - 43.48

The largest possible negative integer n is - 44. We plug it in for n to get 23 x - 44 + 1 = -1011.