Q: when you said you could see that 262147 didn't have a 7 in its factorization, how did you know that? since there is no easy rule for divisibility by 7
A: you can legally knock off the 7 from the right side. and it still will or won't be divisible by 7. do you see what i mean? so, imagine if you could make legal moves like this, until eventually you got it down to something you could recognize. So you can knock the 7 off. then you still have a 14 on the end. so obviously you can subtract 14 and you'll still be left with the same status of 7-ness (or non.) Then you're left with 262, and of course you can divide it all you want and that won't change the 7-ness either. so go ahead and divide it by 2, which is super easy, digitally. just halve the digits and it's 131. Ok so i already know that one is prime just because i know my primes up to at least there. but if you didn't know 131, you can of course subtract 21 (again, that won't change the 7-ness of the main number). so that brings you to 110 and you can drop the zero (again, dividing (by 10) won't change its 7-ness). so you drop the zero and it's 11, and please tell me you can tell by looking at it if 11 is divisible by 7 (it's not). so the original number couldn't have been, because we've only made legal moves up to here