Find the smallest positive integer $N$ for which $N$ is divisible by $19$, and when the digits of $N$ are read in reverse order, the result (after removing any leading zeroes) is divisible by $36$.

[i]Proposed by Michael Tang[/i]
To solve this problem, we need to find the smallest positive integer \( N \) such that:
1. \( N \) is divisible by 19.
2. The reverse of \( N \) is divisible by 36.

Let's break down the problem step by step:

1. **Divisibility by 19**:
   - We need \( N \) to be divisible by 19. Therefore, \( N = 19k \) for some integer \( k \).

2. **Divisibility by 36**:
   - The reverse of \( N \) must be divisible by 36. Since 36 = 4 × 9, the reverse of \( N \) must be divisible by both 4 and 9.
   - For a number to be divisible by 4, its last two digits must form a number that is divisible by 4.
   - For a number to be divisible by 9, the sum of its digits must be divisible by 9.

3. **Finding the smallest \( N \)**:
   - We start by testing multiples of 19 and check if their reverse is divisible by 36.

Let's test the smallest multiples of 19:

- \( 19 \times 1 = 19 \) (reverse: 91, not divisible by 36)
- \( 19 \times 2 = 38 \) (reverse: 83, not divisible by 36)
- \( 19 \times 3 = 57 \) (reverse: 75, not divisible by 36)
- \( 19 \times 4 = 76 \) (reverse: 67, not divisible by 36)
- \( 19 \times 5 = 95 \) (reverse: 59, not divisible by 36)
- \( 19 \times 6 = 114 \) (reverse: 411, not divisible by 36)
- \( 19 \times 7 = 133 \) (reverse: 331, not divisible by 36)
- \( 19 \times 8 = 152 \) (reverse: 251, not divisible by 36)
- \( 19 \times 9 = 171 \) (reverse: 171, not divisible by 36)
- \( 19 \times 10 = 190 \) (reverse: 091, not divisible by 36)
- \( 19 \times 11 = 209 \) (reverse: 902, not divisible by 36)
- \( 19 \times 12 = 228 \) (reverse: 822, not divisible by 36)
- \( 19 \times 13 = 247 \) (reverse: 742, not divisible by 36)
- \( 19 \times 14 = 266 \) (reverse: 662, not divisible by 36)
- \( 19 \times 15 = 285 \) (reverse: 582, not divisible by 36)
- \( 19 \times 16 = 304 \) (reverse: 403, not divisible by 36)
- \( 19 \times 17 = 323 \) (reverse: 323, not divisible by 36)
- \( 19 \times 18 = 342 \) (reverse: 243, not divisible by 36)
- \( 19 \times 19 = 361 \) (reverse: 163, not divisible by 36)
- \( 19 \times 20 = 380 \) (reverse: 083, not divisible by 36)
- \( 19 \times 21 = 399 \) (reverse: 993, not divisible by 36)
- \( 19 \times 22 = 418 \) (reverse: 814, not divisible by 36)
- \( 19 \times 23 = 437 \) (reverse: 734, not divisible by 36)
- \( 19 \times 24 = 456 \) (reverse: 654, not divisible by 36)
- \( 19 \times 25 = 475 \) (reverse: 574, not divisible by 36)
- \( 19 \times 26 = 494 \) (reverse: 494, not divisible by 36)
- \( 19 \times 27 = 513 \) (reverse: 315, not divisible by 36)
- \( 19 \times 28 = 532 \) (reverse: 235, not divisible by 36)
- \( 19 \times 29 = 551 \) (reverse: 155, not divisible by 36)
- \( 19 \times 30 = 570 \) (reverse: 075, not divisible by 36)
- \( 19 \times 31 = 589 \) (reverse: 985, not divisible by 36)
- \( 19 \times 32 = 608 \) (reverse: 806, not divisible by 36)
- \( 19 \times 33 = 627 \) (reverse: 726, not divisible by 36)
- \( 19 \times 34 = 646 \) (reverse: 646, not divisible by 36)
- \( 19 \times 35 = 665 \) (reverse: 566, not divisible by 36)
- \( 19 \times 36 = 684 \) (reverse: 486, not divisible by 36)
- \( 19 \times 37 = 703 \) (reverse: 307, not divisible by 36)
- \( 19 \times 38 = 722 \) (reverse: 227, not divisible by 36)
- \( 19 \times 39 = 741 \) (reverse: 147, not divisible by 36)
- \( 19 \times 40 = 760 \) (reverse: 067, not divisible by 36)
- \( 19 \times 41 = 779 \) (reverse: 977, not divisible by 36)
- \( 19 \times 42 = 798 \) (reverse: 897, not divisible by 36)
- \( 19 \times 43 = 817 \) (reverse: 718, not divisible by 36)
- \( 19 \times 44 = 836 \) (reverse: 638, not divisible by 36)
- \( 19 \times 45 = 855 \) (reverse: 558, not divisible by 36)
- \( 19 \times 46 = 874 \) (reverse: 478, not divisible by 36)
- \( 19 \times 47 = 893 \) (reverse: 398, not divisible by 36)
- \( 19 \times 48 = 912 \) (reverse: 219, not divisible by 36)
- \( 19 \times 49 = 931 \) (reverse: 139, not divisible by 36)
- \( 19 \times 50 = 950 \) (reverse: 059, not divisible by 36)
- \( 19 \times 51 = 969 \) (reverse: 969, not divisible by 36)
- \( 19 \times 52 = 988 \) (reverse: 889, not divisible by 36)
- \( 19 \times 53 = 1007 \) (reverse: 7001, not divisible by 36)
- \( 19 \times 54 = 1026 \) (reverse: 6201, not divisible by 36)
- \( 19 \times 55 = 1045 \) (reverse: 5401, not divisible by 36)
- \( 19 \times 56 = 1064 \) (reverse: 4601, not divisible by 36)
- \( 19 \times 57 = 1083 \) (reverse: 3801, not divisible by 36)
- \( 19 \times 58 = 1102 \) (reverse: 2011, not divisible by 36)
- \( 19 \times 59 = 1121 \) (reverse: 1211, not divisible by 36)
- \( 19 \times 60 = 1140 \) (reverse: 0411, not divisible by 36)
- \( 19 \times 61 = 1159 \) (reverse: 9511, not divisible by 36)
- \( 19 \times 62 = 1178 \) (reverse: 8711, not divisible by 36)
- \( 19 \times 63 = 1197 \) (reverse: 7911, not divisible by 36)
- \( 19 \times 64 = 1216 \) (reverse: 6121, not divisible by 36)
- \( 19 \times 65 = 1235 \) (reverse: 5321, not divisible by 36)
- \( 19 \times 66 = 1254 \) (reverse: 4521, not divisible by 36)
- \( 19 \times 67 = 1273 \) (reverse: 3721, not divisible by 36)
- \( 19 \times 68 = 1292 \) (reverse: 2921, not divisible by 36)
- \( 19 \times 69 = 1311 \) (reverse: 1131, not divisible by 36)
- \( 19 \times 70 = 1330 \) (reverse: 0331, not divisible by 36)
- \( 19 \times 71 = 1349 \) (reverse: 9431, not divisible by 36)
- \( 19 \times 72 = 1368 \) (reverse: 8631, not divisible by 36)
- \( 19 \times 73 = 1387 \) (reverse: 7831, not divisible by 36)
- \( 19 \times 74 = 1406 \) (reverse: 6041, not divisible by 36)
- \( 19 \times 75 = 1425 \) (reverse: 5241, not divisible by 36)
- \( 19 \times 76 = 1444 \) (reverse: 4441, not divisible by 36)
- \( 19 \times 77 = 1463 \) (reverse: 3641, not divisible by 36)
- \( 19 \times 78 = 1482 \) (reverse: 2841, not divisible by 36)
- \( 19 \times 79 = 1501 \) (reverse: 1051, not divisible by 36)
- \( 19 \times 80 = 1520 \) (reverse: 0251, not divisible by 36)
- \( 19 \times 81 = 1539 \) (reverse: 9351, not divisible by 36)
- \( 19 \times 82 = 1558 \) (reverse: 8551, not divisible by 36)
- \( 19 \times 83 = 1577 \) (reverse: 7751, not divisible by 36)
- \( 19 \times 84 = 1596 \) (reverse: 6951, not divisible by 36)
- \( 19 \times 85 = 1615 \) (reverse: 5161, not divisible by 36)
- \( 19 \times 86 = 1634 \) (reverse: 4361, not divisible by 36)
- \( 19 \times 87 = 1653 \) (reverse: 3561, not divisible by 36)
- \( 19 \times 88 = 1672 \) (reverse: 2761, not divisible by 36)
- \( 19 \times 89 = 1691 \) (reverse: 1961, not divisible by 36)
- \( 19 \times 90 = 1710 \) (reverse: 0171, not divisible by 36)
- \( 19 \times 91 = 1729 \) (reverse: 9271, not divisible by 36)
- \( 19 \times 92 = 1748 \) (reverse: 8471, not divisible by 36)
- \( 19 \times 93 = 1767 \) (reverse: 7671, not divisible by 36)
- \( 19 \times 94 = 1786 \) (reverse: 6871, not divisible by 36)
- \( 19 \times 95 = 1805 \) (reverse: 5081, not divisible by 36)
- \( 19 \times 96 = 1824 \) (reverse: 4281, not divisible by 36)
- \( 19 \times 97 = 1843 \) (reverse: 3481, not divisible by 36)
- \( 19 \times 98 = 1862 \) (reverse: 2681, not divisible by 36)
- \( 19 \times 99 = 1881 \) (reverse: 1881, not divisible by 36)
- \( 19 \times 100 = 1900 \) (reverse: 0091, not divisible by 36)
- \( 19 \times 101 = 1919 \) (reverse: 9191, not divisible by 36)
- \( 19 \times 102 = 1938 \) (reverse: 8391, not divisible by 36)
- \( 19 \times 103 = 1957 \) (reverse: 7591, not divisible by 36)
- \( 19 \times 104 = 1976 \) (reverse: 6791, not divisible by 36)
- \( 19 \times 105 = 1995 \) (reverse: 5991, not divisible by 36)
- \( 19 \times 106 = 2014 \) (reverse: 4102, not divisible by 36)
- \( 19 \times 107 = 2033 \) (reverse: 3302, not divisible by 36)
- \( 19 \times 108 = 2052 \) (reverse: 2502, not divisible by 36)
- \( 19 \times 109 = 2071 \) (reverse: 1702, not divisible by 36)
- \( 19 \times 110 = 2090 \) (reverse: 0902, not divisible by 36)
- \( 19 \times 111 = 2109 \) (reverse: 9012, not divisible by 36)
- \( 19 \times 112 = 2128 \) (reverse: 8212, not divisible by 36)
- \( 19 \times 113 = 2147 \) (reverse: 7412, not divisible by 36)
- \( 19 \times 114 = 2166 \) (reverse: 6612, not divisible by 36)
- \( 19 \times 115 = 2185 \) (reverse: 5812, not divisible by 36)
- \( 19 \times 116 = 2204 \) (reverse: 4022, not divisible by 36)
- \( 19 \times 117 = 2223 \) (reverse: 3222, not divisible by 36)
- \( 19 \times 118 = 2242 \) (reverse: 2422, not divisible by 36)
- \( 19 \times 119 = 2261 \) (reverse: 1622, not divisible by 36)
- \( 19 \times 120 = 2280 \) (reverse: 0822, not divisible by 36)
- \( 19 \times 121 = 2299 \) (reverse: 9922, not divisible by 36)
- \( 19 \times 122 = 2318 \) (reverse: 8132, not divisible by 36)
- \( 19 \times 123 = 2337 \) (reverse: 7332, not divisible by 36)
- \( 19 \times 124 = 2356 \) (reverse: 6532, not divisible by 36)
- \( 19 \times 125 = 2375 \) (reverse: 5732, not divisible by 36)
- \( 19 \times 126 = 2394 \) (reverse: 4932, divisible by 36)

After testing the multiples of 19, we find that \( N = 2394 \) is the smallest number that satisfies both conditions.

The final answer is \( \boxed{2394} \).