Interesting divisibility problem

Question

Several different numbers are written on the board. Exactly 2 are divisible by 2, and exactly 33 are divisible by 33. Let M be the biggest of them. What is the smallest possible value for M?

Answer 1

The problem is to find the smallest number M that is greater than or equal to 33 multiples of 33 (but not multiple of 2) and 2 multiples of 2 (but not multiple of 3, since it said "exactly").

The 33 smallest multiples of 33 (but not multiple of 2) are:

count number

1        33

         2        99

         3        165

         4        231

         5        297

         6        363

         7        429

         8        495

         9        561

         10        627

         11        693

         12        759

         13        825

         14        891

         15        957

         16        1023

         17        1089

         18        1155

         19        1221

         20        1287

         21        1353

         22        1419

         23        1485

         24        1551

         25        1617

         26        1683

         27        1749

         28        1815

         29        1881

         30        1947

         31        2013

         32        2079

         33        2145

Now, we only need 2 even numbers (not in the above list) that are less than M=2145.

Use 100 and 102.

The smallest possible value of M is 2145.

Answer 2

The two divisible by 2 are 66 and 132. The remaining numbers the odd multiples of 33.

The 31st odd number is 61 (2n - 1).

(31)(61) = ?