You've just been handed the mathematical red pill! That bizarre pattern where 111, 222, 333... through 999 are all divisible by 37 but not by 11 is one of those mathematical quirks that makes your brain do a double-take.
The secret? Every repeating 3-digit number (like 777) equals 9×9×n, where n is the repeating digit. So 777 = 9×9×7 = 81×7 = 567+210 = 777. And guess what? 81 is divisible by 27! But 11 doesn't divide these numbers because... well, 11 is just too mainstream for this mathematical matrix glitch.
Next time someone shows you this, just slowly remove your sunglasses and whisper "I know number theory."