$2^{29}$ has $9$ distinct digits. Find the missing digit without the use of a calculator.
I've seen its solution before but I still don't understand it. Math novice here. A detailed answer will be appreciated. Thanks
The number $2^{29}$ has $9$ distinct digits. Find the missing digit without the use of a calculator.
Asked
Active
Viewed 1,756 times
1
Paradox 101
- 1,091
-
@user291957 - I think that if the poster knew the answer to that question, then they would not be asking it here! – Simon Rose Sep 22 '14 at 07:47
-
I misunderstood the question. Deleting the comment right now @Simon Rose – Jasser Sep 22 '14 at 07:53
-
are you familiar with congruences/modular arithmetic @Paradox101 – AgentS Sep 22 '14 at 08:01
-
2Check http://math.stackexchange.com/questions/819412/the-number-229-has-exactly-9-distinct-digits-which-digit-is-missing – Macavity Sep 22 '14 at 08:45
-
yes i'm somewhat familiar with congruences – Paradox 101 Sep 22 '14 at 10:25
-
@Macavity I've seen that link before, but i don't really get it. Why should we take 2^29 mod 9? – Paradox 101 Sep 22 '14 at 10:27
-
@Macavity a little more elaboration needed. Thanks – Paradox 101 Sep 22 '14 at 11:27
-
10^k = 1 (mod 9) – Paradox 101 Sep 22 '14 at 11:32
-
$10 = 1 \pmod 9 \implies 10\times 10 = 1\times 1 = 1 \pmod 9$ etc. So $10^k = 1^k = 1 \pmod 9$. $$ $$ Alternately, note $10^k = 9999..999 + 1$ (where there are k-1 $9$s). – Macavity Sep 22 '14 at 11:34
1 Answers
2
You have $9$ distinct digits, one is missing from among $0$ to $9$. The sum of all should give you $45$ (why?). So if you had the sum of all digits, you could easily determine the missing one.
The next best thing is taking the number $\pmod 9$ - which gives you the sum of digits $\pmod 9$ as $10^k = 1 \pmod 9$. To show this, let $2^{29} = \sum_{k=0}^9 a_k 10^k$. Then $$2^{29} \equiv \sum_{k=0}^9a_k 10^k \equiv \sum_{k=0}^9a_k\pmod 9 $$
So in this case, this remainder gives you the answer, as you're looking for a one digit solution.
P.S. - This is a duplicate, so perhaps after studying it you could remove the Q.
Macavity
- 46,381
-
1Clever method; works for all nine digit numbers with no duplicate digits unless the missing digit is 0 or 9. – gnasher729 Sep 22 '14 at 12:03