Determine all possible integers n less than 10¹⁰⁰ that simultaneously satisfy these conditions:

• n divides 2ⁿ
• n - 1 divides 2ⁿ - 1
• n - 2 divides 2ⁿ - 2

Note: As an extra challenge, determine a semi-analytic (simple calculator + p&p) solution to this problem.

**** Adapted from a problem appearing at a William Lowell Putnam Mathematics Competition