To solve the problem of finding the smallest positive integer (n) such that when 120 is divided by (n) the remainder is 12, follow these steps:

Step 1: Apply the division algorithm

When 120 is divided by (n) with remainder 12, we have:
(120 = qn + 12) (where (q) is the quotient, and remainder < divr: (12 < n)).

Rearranging gives:
(qn = 120 - 12 = 108).

Step 2: Find divrs of 108 greater than 12

The divrs of 108 are: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108.

We need divrs >12: 18, 27, 36, 54, 108.

Step 3: Select the smallest divr

The smallest among these is 18.

Answer: (\boxed{18})

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