Here are the smallest values for N I could find:

**121** has 3 divisors : 1, 11, 121

All divisors of 121 contain the digit **1**

**254** has 4 divisors: 1, 2, 127, 254

Divisors (2, 127, 254) contain the digit **2**

**39** has 4 divisors: 1, 3, 13, 39

Divisors (3, 13, 39) contain the digit **3**

**1849** has 3 divisors: 1, 43, 1849

All divisors of 1849 (with the exclusion of 1) contain the digit **4**

**25** has 3 divisors: 1, 5, 25

Divisors (5, 25) contain the digit **5**

**16043** has 4 divisors: 1, 61, 263, 16043

All divisors of 16043 (with the exclusion of 1) contain the digit **6**

**497** has 4 divisors: 1, 7, 71, 497

Divisors (7, 71, 497) contain the digit **7**

**6889** has 3 divisors: 1, 83, 6889

All divisors of 6889 (with the exclusion of 1) contain the digit **8**

**1691** has 4 divisors: 1, 19, 89, 1691

Divisors (19, 89, 1691) contain the digit **9**

Double digits:

**10201** has 3 divisors: 1, 101, 10201

Divisors (101, 10201) contain **10**

**171121** has 4 divisors: 1, 211, 811, 171121

Divisors (211, 811, 171121) contain **11**

**16129** has 3 divisors: 1, 127, 16129

Divisors (127, 16129) contain **12**

**Other results:**

**Digit 1 :**

**1859** : 1, 11, 13, 143, 169, 1859

**Digit 2 :**

**32258** : 1, 2, 127, 254, 16129, 32258

**Digit 3 :**

**34917** : 1, 3, 103, 113, 309, 339, 11639, 34917

**Digit 4 :**

**Digit 5 :**

**75815** : 1, 5, 59, 257, 295, 1285, 15163, 75815

**Digit 6 :**

**Digit 7 :**

**5767** : 1, 73, 79, 5767

**53179** : 1, 7, 71, 107, 497, 749, 7597, 53179

**Digit 8 :**

**7387** : 1, 83, 89, 7387

**Digit 9 :**

**150499** : 1, 19, 89, 1691, 7921, 150499