Sum of digits divisible by $27$

Question

I know that every third number is divisible by $3$ and hence, sum of its digits is divisible by $3$. Same holds for $9$ also. But how do we generalise it? We know that the divisibility condition for higher powers of $3$ is not about the sum of digits. How can we find $n$ such that in a group of $n$ consecutive positive integers, there is a number such that the sum of its digits is divisible by $27$ (or $81,$ say)? Does it exist? Please prove or disprove.

Answer 1

New answer instead of editing the - already accepted - answer because the argument is significantly different (and much simpler).

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Let $Q(x)$ denote the digit sum of $x$.

Let $r\ge 1$. Then $n=10^r-1=\underbrace{99\ldots 9}_r$ is the smallest $n$ such that among any $n$ consecutive positive integers, at least one has digit sum a multiple of $9r$.

That no smaller $n$ works, is immediately clear because in $1,2,3,\ldots, 10^r-2$, all digit sums are $>0$ and $<9r$.

Remains to show that in any sequence of $n$ consecutive integers, a digit sum divisible by $9r$ occurs. This is well-known for $r=1$. For $r>1$, consider $n$ consecutive positive integers $$a,a+1,\ldots, a+n. $$ Among the first $9\cdot 10^{r-1}=n-(10^{r-1}-1)$ terms, one is a multiple of $9\cdot 10^{r-1}$. Say, $9\cdot 10^{r-1}\mid a+k=:b$ with $0\le k<9\cdot 10^{r-1}$. Then $Q(b)$ is a multiple of $9$, and as the lower $r-1$ digits of $b$ are zero, we have $Q(b+i)=Q(b)+Q(i)$ for $0\le i<10^{r-1}$ and hence $$Q( b+10^j-1)=Q(b)+9j,\qquad 0\le j\le r-1.$$ (Note that $k+10^{r-1}-1<10^r-1$, so these terms are really all in our given sequence). It follows that $9r$ divides one of these $Q(b+10^j-1)$.

Answer 2

Note that the natural numbers $\{1,2,\cdots, 999\}$ contain integers for which the sum of the digits is any specified value $\pmod {27}$

Considering the multiples of $1000$ we see that each block of $1000$ integers contains one which ends in three $0's$.

Starting from any integer $k$, we go to the next multiple of $1000$ (a gap of at most $999$). We then add whatever three (or fewer) digit integer we need to "correct" the sum of the digits $\pmod {27}$, which takes, at most, another $999$.

Thus, every block of $2\times 999$ consecutive integers contains at least one for which the sum of the digits is a multiple of $27$. A similar argument works for any desired divisor.

I expect the bound could be tightened considerably, but at least this shows that a bound exists.

Answer 3

Let $Q(x)$ denote the digit sum of $x$.

$999$ is the smallest $n$ such that among any $n$ consecutive positive integers, at least one has digit sum a multiple of $27$.

First note that none of the $998$ consecutive integers $1,2,\ldots ,998$ has digit sum a multiple of $27$.

Any sequence of $999$ consecutive integers is either of the form $$\tag11000N+1,\ldots, 1000N+999$$ or $$\tag21000N+k+2,\ldots, 1000N+999,1000(N+1),\ldots, 1000(N+1)+k$$ with $0\le k\le 997$.

In $(1)$, the digit sums are $Q(N)+Q(i)$ with $i$ running from $1$ to $999$ and hence $Q(i)$ covering all values from $1$ to $27$. We conclude that $(1)$ contains a term with digit sum a multiple of $27$.

So let's look at $(2)$: We know $Q(N+1)\equiv Q(N)+1\pmod 9$, hence $Q(N+1)\equiv Q(N)+(1\text{ or }10\text{ or }19)\pmod{27}$.

• If $Q(N+1)\equiv 0\pmod{27}$, then already $1000(N+1)$ has the desired property.

• If $Q(N+1)\equiv 1\pmod {27}$, then among $1000(N+1),\ldots, 1000(N+1)+899$, all remainders $\bmod27$ occur, which solves the problem for all $k\ge 899$. For $k\le 898$, the sequence contains $1000N+900$, $1000N+909$, and $1000N+999$ with digit sums $Q(N)+9$, $Q(N)+18$, $Q(N)+27$. As $Q(N)\bmod 27$ is one of $0$, $9$, $18$, we are done.

• More generally, if $Q(N+1)\equiv r\pmod {27}$ with $1\le r\le 9$, then $Q(1000(N+1)+999-100r)=Q(N+1)+27-r\equiv 0\pmod{27}$, which solves the problem for all $k\ge 999-100r$. For $k\le 998-100r$, the sequence contains $1000N+(1000-100r)$, $1000N+(1009-100r)$, and $1000N+(1099-100r)$ with digit sums $Q(N)+10-r$, $Q(N)+19-r$, $Q(N)+28-r$. As $Q(N)\bmod 27$ is one of $r-1$, $r+8$, $r+17$, we are done.

• If $Q(N+1)\equiv 10+r\pmod{27}$ with $0\le r\le 8$, then $Q(1000(N+1)+89-10r)=Q(N+1)+17-r\equiv 0\pmod{27}$, which solves the problem for all $k\ge 89-10r$. For $k\le 89-10r$, the sequence contains $1000N+999-r$, $1000N+909-r$, and $1000N+900-100r$ with digit sums $Q(N)+27-r$, $Q(N)+18-r$, $Q(N)+9-r$. As $Q(N)\bmod 27$ is one of $r$, $r+9$, $r+18$, we are done.

• If $Q(N+1)\equiv 19+r\pmod{27}$ with $0\le r\le 7$, then $Q(1000(N+1)+8-r)=Q(N+1)+8-r\equiv 0\pmod{27}$, which solves the problem for all $k\ge 8-r$. For $k\le 8-r$, the sequence contains $1000N+999-r$, $1000N+909-r$, and $1000N+900-100r$ with digit sums $Q(N)+27-r$, $Q(N)+18-r$, $Q(N)+9-r$. As $Q(N)\bmod 27$ is one of $r$, $r+9$, $r+18$, we are done.

Answer 4

Obviously there are number whose digits add to $27$. ($999$ or $524385$ etc.) and obviously the sum of the digits $27$ is a multiple of $9$ so they are a multiple of $9$ but are they a multiples of $27$; and must multiples of $27$ have digits adding to a multiple of $27$.

Well, $27$ itself is an obvious counter example of the latter.

And $999= 27*37$ but $524385= 27*19421\frac 23$ so the first is not true either.

So the question I guess is why not?

Well the rule works for $9$ because $9 = 10-1$. And it works for $3$ because $3|9$.

Details: If $k|b-1$ and $n= \sum_{i=0}^m a_ib^i= \sum_{i=0}^m (a_i)(b^i-1) + \sum_{i=0}^m a_i$. Now $b^i-1 =(b-1)(b^{i-1} + b^{i-2}+ ..... + 1)$ so $(b-1)$ divides all of the $b^i-1$ so $b-1$ divides $n$ if and only if $(b-1)$ divides $\sum_{i=0}^m a_i$. If we let $b= 10$ and $b-1=9$ and $a_i$ be the digits of $n$ that's our result.

And it follows that if $k|b-1$ then $k|(b^i-1)$ so $k|n$ if and only if $k$ divides $\sum_{i=0}^m a_i$ as well.

And this will be true for any decimal system base $b$ (not just $b=10$ and and $k|b-1$ (not just $3|9$).

The fact that $3^2 = 9$ is mostly a coincidence and powers of $3$ is a bit of a red herring. It's not powers of $3$ going up that matter, but factors $10-1$ going down that matter.

We can note that in base $7$ a number is a multiple of $6$ if and only if the sum of the digits is a multiple of $6$ and a multiple of $2$ or of $3$ if and only if the sum of the digits is a multiple of $2$ or of $3$ respectively but nothing can be said of $4$ or $3$. ($11_7 = 8$ is a multiple of $4$ but $1+1=2$ is not. And $12_7 =9$ is a multiple of $9$ but $1+2=3$ which is not.)

Why doesn't it work? Well. $27 = 3*(10-1)= (3-1)*10 + (10-3)$. The sum of the digits of $27$ are $(3-1) + (10-3) = 10-1$. Our rule of $9$s apply and we can't jump magically to $27$. And if we increas by $27$ if we ignore carrying and borrowing we get $ab + 27 = (a+2)(b+7)$ and the sums of the digits are $a+b + 9$. That's an increase of $9$; not of $27$. Ind if we carry (i.e. $b \ge 3$ or $a \ge 8$ or $b\ge 3$ and $a\ge 7$) we get the sums are $(a+2+1)(b+7-10)$ or $1(a+2-10)(b+7)$ or $1(a+2+1-10)(b+7 - 10)$ and the sum of the digits stay the same or decreases by $9$; not $27$.

But if $b-1 = k^m$ then in base $b$ we will have that multiples of $k^i; i\le m$ will have the sum of the digits add to a multiple of $k^i$.

Example in base $28$ then sum of the digits of a multiple of $27$ will add to a multiple of $27$.

FWIW $999_{10} = 28^2 + 7*28 + 19 = 17T_{28}$ where $T$ is the digit for $19$.

And $27*92 = 2484_{10} = 3*28^2+4*28 + 20= 34U_{28}$ where $U$ is the digit for $20$ is another example.

Less trivial example. The number $8ATR_{28}$ were $A=10$ and $T=19$ and $R=17$ will have digits that add to $8+10+19+17=54$, so my claim is that it ought to be a multiple of $27$. And

$8ATR_{28} = 8*28^3 + 10*28^2+ 19*28 + 17 =$

$8(27+ 1)^3 + 10(27+1)^2 + 19(27+1) + 17 =$

$8(27^3 + 3*27^2+3*27 + 1) + 10(27^2 + 2*27 + 1) + 19(27+1)+17=$

$[8*27^3 + 3*27^2 + 3*27 + 20*27^2 + 2*27 + 19*27] + 8 + 10 + 19+17=$

$27(8*27^2 + 3*27 + 3 + 20*27 + 2 + 19) + 54 = $

$27(8*27^2 + 3*27 + 3 + 20*27 + 2 + 19 + 2)$ is a multiple of $27$.

And indeed $8*28^3 + 10*28^2+ 19*28 + 17=184005 = 27*6815$