Find the Missing Digit in 229
We know 229 is a 9-digit number containing every digit exactly once, except for one missing digit. We do not need to compute 229.
Step 1 · Use the digit sum
A number has the same remainder modulo 9 as the sum of its digits.
0
1
2
3
4
5
6
7
8
9
0 + 1 + 2 + ··· + 9 = 45
If the missing digit is d, then the digit sum is:
45 − d
Step 2 · Compute only 2²⁹ mod 9
Powers of 2 modulo 9 repeat:
2¹ mod 92
2² mod 94
2³ mod 98
2⁴ mod 97
2⁵ mod 95
2⁶ mod 91
Since 29 = 6 × 4 + 5,
229 ≡ 25 ≡ 5 (mod 9)
Step 3 · Match the remainders
45 − d ≡ 5 (mod 9)
↓
−d ≡ 5 (mod 9)
↓
d ≡ 4 (mod 9)
Since d must be a digit from 0 to 9:
4
is the only possibility.