The modulus validation

Jelly, 7 bytes

%⁼⁴
0ç#

This is a full program. Arguments are divisors, target moduli, and number of solutions, in that order.

Try it online!

How it works

0ç#  Main link.
     Left argument: D (array of divisors)
     Right argument: M (array of target moduli)
     Third argument: n (number of solutions)

0ç#  Execute the helper link with k = 0, 1, 2, ... as left argument and D as the
     right one until n of them return 1. Yield the array of matches.


%⁼⁴  Helper link. Left argument: k. Right argument: D

%    Compute k % d for each d in D.
 ⁼⁴  Compare the result with M.

JavaScript (ES6), 74 70 69 bytes

Takes input as an integer n and an array a of [modulo, remainder] arrays with currying syntax (n)(a).

n=>a=>eval('for(i=r=[];a.some(([b,c])=>i%b-c)||--n*r.push(i);i++);r')

Test cases

let f =

n=>a=>eval('for(i=r=[];a.some(([b,c])=>i%b-c)||--n*r.push(i);i++);r')

console.log(f(5)([[3, 2], [4, 1], [5, 3]]))
console.log(f(3)([[8, 0], [13, 3], [14, 8]]))

Perl 6, 33 bytes

{grep((*X%@^b)eqv@^c,0..*)[^$^a]}

Try it

The input is ( number-of-values, list-of-divisors, list-of-remainders )

Expanded:

{   # bare block lambda with placeholder parameters ｢$a｣ ｢@b｣ ｢@c｣

  grep(

    # WhateverCode lambda:
    (

      *        # the value being tested

      X%       # cross modulus

      @^b      # with the divisors ( second parameter )

    )

    eqv        # is that list equivalent with

    @^c        # the expected remainders ( third parameter )

    # end of WhateverCode lambda

    ,

    0 .. *     # Range of all Integers starting with 0

  )[ ^$^a ]    # grab up-to ｢$a｣ values ( first parameter )
               # ( ｢^$a｣ is the same as ｢0 ..^ $a｣ )
}