Couldn't we do the 3 mod 5 and 5 mod 7 first and notice that they are both -2, and therefore those conditions together are equal to -2 mod 35 or 33 mod 35, and then include the 2 mod 3 in to try the results? You only have to try 2 numbers, 33 and 68 (because 68 is 2 mod 3)...
@Potato2017 Yes, that's a great way to do this problem! I like your idea. It is faster than the current explanation, and I'll try to incorporate it into the lesson. Thanks very much for sharing!
@Potato2017 That's pretty cool!