Question

@whpalmer4 asked an interesting question.
If there is only one 3-digit number in the base \(b\) the sum of the numbers of which is \(b+1\) times less then the number.

Answer 1

here's what we noticed:

There is only one such number in base \(b\) such that its sum of the number's digits is \(\dfrac{1}{b + 1}\) of the number.

Answer 2

We want to prove if that is true. :-)

Answer 3

For example, in base \(10\), there is only one such number such that the sum of its digits is \(\dfrac{1}{10 + 1}\) the number: 198.

In base 16, he told us that there is only one such number which has its sum of digits equal to \(\dfrac{1}{17}\) the number.

Answer 4

Can we prove this?

Answer 5

This is the beginning of this problem. @ParthKohli solved it. I wrote a code in Mathematica and got this.
\(b=2\quad n=6=110_2\)
\(b=3\quad n=16=121_3\)
\(b=4\quad n=30=132_4\)
\(b=5\quad n=48=143_5\)
\(b=6\quad n=70=154_6\)
\(b=7\quad n=96=165_7\)
\(b=8\quad n=126=176_8\)
\(b=9\quad n=160=187_9\)
\(b=10\quad n=198=198_{10}\)
\(b=11\quad n=240=\text{1a9}_{11}\)
\(b=12\quad n=286=\text{1ba}_{12}\)
\(b=13\quad n=336=\text{1cb}_{13}\)
\(b=14\quad n=390=\text{1dc}_{14}\)
\(b=15\quad n=448=\text{1ed}_{15}\)
\(b=16\quad n=510=\text{1fe}_{16}\)
For any base there is only one number.

Answer 6

Yes, we did it!

Answer 7

We have completed a discovery!

Answer 8

Can we expand this discovery outside three-digit numbers?

Answer 9

Proof:
Any 3-digit number in base \(b\) can be given as \(a_2b^2+a_1b+a_0\). Now divide this number by \(b+1\):
\(a_2b^2+a_1b+a_0=(a_2b+(a_1+a_2))(b+1)+(a_0-a_1+a_2)\)
If we want this number to be divisible by \(b+1\) we need
\(a_0-a_1+a_2=0\).
Next one is the statement of the problem:
\((b+1)(a_2+a_1+a_0)=a_2b^2+a_1b+a_0\)
Change this a little:
\(a_2(b^2-b-1)-a_1-a_0b=0\)
As we know from the previous \(a_1=a_0+a_2\)
Put this into the last expression:
\(a_2(b^2-b-2)-a_0(b+1)=0\)
Can divide by \(b+1\):
\(a_2(b-2)-a_0=0\)
That means that \(a_0\) must be divisible by \(b-2\). But \(a_0\) can take only values \(0,1,\ldots,b-1\). Among them only \(b-2\) is divisible by \(b-2\). So
\(a_0=b-2\)
Using the last 2 equations
\(a_2=1\)
And then
\(a_1=b-1\)
The answer:
Only one number exists and it is \(n=\overline{1(b-1)(b-2)}\).
That was 198 for b=10. Check this for other b.

Answer 10

@Preetha Look what we achieved! Thanks to @klimenkov :-)

Answer 11

@ParthKohli, @whpalmer4
How can we expand our problem on non 3-digit numbers?

Answer 12

This is awesome. I can't say I understand it, but it is a great example of collaborative learning, where together you achieved something. Thanks so much for sharing.

Answer 13

nice :).. however i was wondering about one of your statements, wat about the assumption that ao-a1+a1=n(b+1)

Answer 14

There is a theorem in algebra about dividing polynomials. We can divide polynomials with the reminder. And the degree of the reminder must be lower than the degree of the divider. So it will never happen.

Answer 15

i did not know its such an old post! wow but still pretty awesome

Answer 16

you have any new proofs?

Answer 17

Yes, some day I will post some of them.

Answer 18

everything makes sense but just 1 more question again if you dont mind xD you said the degree must lower than the divider in this polynomial division but what if a0-a1+a2 just so happens to be b-1

Answer 19

and cool:) i look forward to it

Answer 20

is that also not possible being equal?

Answer 21

Yes, because \(b-1\) has degree 1. It has to be lower than 1, only 0 degree will be okay.

Answer 22

ok just making sure thnx