The LWE problem is one level of abstraction away from the fundamental lattice problems it reduces to. It is somewhat analogous to the Diffie-Hellman problem that many constructions reduce to, which itself is related to the lower-level discrete logarithm problem.

The lattice equivalent of integer factorization is the shortest vector problem: you're given n vectors of length m, and you have to find the sum of integer multiples of those vectors that comes closest to (or a small factor away from the closest) the zero vector. Say you have the 4 vectors

    [ 3 92  4  2]
    [54  0 92 41]
    [19 91 61 48]
    [39 59 40 14].

The shortest vector that you can obtain from adding integer multiples of these vectors is [19 -8 -15 2], which you can obtain by 3*[39 59 40 14] + [19 91 61 48] - 2*[54 0 92 41] - 3*[ 3 92 4 2].

With only 4 vectors it is easy to find the solution here. But the hardness grows exponentially with the dimension, and the dimensions in cryptographically relevant lattices are in the hundreds to thousands.