I’m thinking of a 4-digit number. If I add its four digits together I get 4. Further, the 4-digit number is divisible by 55. What numbers could I possibly be thinking of?
Solution
(i) It’s first important to establish what the four digits could be. What four digits when added together give a total of 4?
Clearly there may be zeros in this number I’m thinking of. So let me see what four numbers including zero, add to 4. I’ll do this systematically, by thinking of the non-zero digits.
No number is bigger than 4.
- If there is a 4 all of the rest of the numbers are zero.
- If there is a 3 (and therefore no 4s) it must go with a 1. So we have 3, 1, 0, 0.
- If there is a 2 (and therefore no 3s or 4s) it must go with a 2 or two 1s. So we have 2, 2, 0, 0 or 2, 1, 1, 0.
- If there are no 2s, 3s or 4s there must be four 1s. So we have 1, 1, 1, 1.
(ii) So what numbers can be made up with the restricted sets above and be divisible by 55? First let’s note that if a number is divisible by 55 it is divisible by 5 (so it ends in 0 or 5) and 11.
I’ll use the numbered sets from (i).
- One 4 and five zeros gives only the number 4000. (0400 is not allowed as it is not, strictly speaking, a 4-digit number.) This is not divisible by 11.
- The numbers here can only be 3100, 3010, 3001, 1300, 1030, and 1003. None of these are divisible by 11.
- Here 2200 and 1210 pass the tests.
- The only number here, 1111 is divisible by 11 but not by 5. The only possible numbers are 2200 and 1210.
Extension
Make up some problems like this to try on your friends. You could start off with 2- or 3-digits numbers.