How can you quickly judge whether a base-conversion claim is impossible?

"1. If I told you that the decimal number (i.e., base-10 number) 2022 was equal to 13621₆, would you call me a liar without even having to think too hard? Yes, you should. A number in base-6 can’t have any digits in it other than 0 through 5, and the “number” I tried to give you had a 6 in it.\n\n2. If I told you that the decimal number 2022 was equal to 1413₆, would you call me a liar without even having to think too hard? Yes, you should. Think about it: in base 6, each digit’s place value (except the one’s place) is worth less than it is in base 10. Instead of a ten’s place, hundred’s place, and thousand’s place, we have a woosy six’s place, thirty-six’s place, and two-hundred-and-sixteen’s place. So there’s no way that a number whose base-6 digits are 1, 4, 1, and 3 would be as large as a number whose base-10 digits are 2, something, something, and something. Put another way, if the base is smaller, the number itself has to “look bigger” to have a chance of evening that out.\n\n3. If I told you that the decimal number 2022 was equal to 8FA₁₆, would you call me a liar without even having to think too hard? Yes, you should, because of the mirror reflection of the above logic. Every digit of a hexadecimal number (again, except the one’s place) is worth more than it is in base 10. So a four-digit hex number beginning with an 8 is going to be way bigger than a wimpy four-digit decimal number beginning with 2."