2 ms·

Can someone ELI5 what the issue is with numbers that have a remainder of 4 or 5 after being divided by 9? I'm assuming it's something simple that I'm just not s

• kens 8y ago
  The issue is that each term n^3 equals +1, 0 or -1 mod 9. (Test the numbers 0-9). When you add three of these values, you can't get 4 or 5 (mod 9). Thus, you can't get a result that equals 4 or 5 mod 9.
• OscarCunningham 8y ago
  The only possible values of a cube mod 9 are 0, 1 and 8. (If you expand (9n+r)^3 you'll see that all the terms divide by 9 except r^3. So the value of a cube mod 9 depends only on its own mod 9 value. So we only need to check that 0, 1 and 8 are the only values achieved by the first 9 cubes.) If you add three of these values the only possible answers are 0, 1, 2, 3, 6, 7 and 8 mod 9.