Linked Questions

3
votes
2answers
643 views

Proving that $9$ is a divisor of $x \in \Bbb N$ if the sum of digits of $x$ is divisible by $9$. [duplicate]

Suppose x is a positive integer with $n$ digits, say $x = d_1d_2d_3\ldots d_n.$ If $9$ is a divisor of $d_1 + d_2 + \ldots d_n$, prove then $9$ is a divisor of $x$. My attempt: suppose $x = 4518.$ ...
0
votes
3answers
48 views

proof that the sum of digits of natural number are divisible by 3 iff the number is [duplicate]

Im trying to prove that every natural number is divisble by three if and only if the sum of its digits are divisible by three. First i proved by induction that $10^n-1$ is divisible by 9 (and ...
0
votes
1answer
48 views

If $3$ divides the decimal digit sum of $n$ then $3$ divides $n$ (casting out threes) [duplicate]

This is a trick I learnt in primary school, but never gave it much thought. Here's how I formulate it: $$ n = \sum_{j=0}^{m} x_j 10^{m-j} $$ is a decimal expansion of some integer $n$ such that $$ \...
0
votes
3answers
52 views

Looking for a theorem talking about the remainder when a number is divided by 9 [duplicate]

I have come across an interesting property of the number 9, which some people call it casting out nines. This is the property: If any number is divisible by 9, then you can keep adding the digits ...
-1
votes
1answer
29 views

Prove $ n \equiv s(n)\ ($mod$\ 3)$ using the fact that $\ [10^n] = [1]$. [duplicate]

Prove $ n \equiv s(n)\ (mod\ 3)$ using the fact that $\ [10^n] = [1]$. Let $n = (a_k \times 10^k) + (a_{k-1} \times 10^{k-1}) + \cdots +(a_1 \times 10^1)+ (a_0 \times 10^0)$ and $s(n)=(a_k + a_{k-1}+ \...
0
votes
0answers
25 views

Prove that (integer)-(the sum of it's digits) can be divided by 9 [duplicate]

How to prove this: Choose whichever integer you like Subtract from it the sum of it's digits The result can always be divided by 9 For example: I choose 123. The sum of it's digits is 1+2+3=6. ...
81
votes
9answers
4k views

Divisibility by 7 rule, and Congruence Arithmetic Laws

I have seen other criteria for divisibility by 7. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. 323) is interesting, but could not prove it. Let $n = (...
31
votes
7answers
6k views

Do odd imaginary numbers exist?

Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
14
votes
4answers
3k views

How can I tell if a number in base 5 is divisible by 3?

I know of the sum of digits divisible by 3 method, but it seems to not be working for base 5. How can I check if number in base 5 is divisible by 3 without ...
6
votes
7answers
1k views

Prove $10^{n+1}+3\cdot 10^n+5$ is divisible by $9$?

How do I prove that an integer of the form $10^{n+1}+3\cdot 10^{n}+5$ is divisible by $9$ for $n\geq 1$?I tried proving it by induction and could prove it for the Base case n=1. But got stuck while ...
7
votes
3answers
3k views

Ways to check whether a number is multiple of another number.

We know that, giving a number, by adding up each of its digit, and mod the result by 3, if the reminder is 0, then the number is a multiple of 3, otherwise, it's not. This algorithm works for ...
2
votes
3answers
4k views

Rules of Division

I know a few rules number ends with even digit, it is divisible by 2 number ends with 5 or 0 is divisible by 5 if sum of all digits in a number is divisible by 3 then that number is divisible by 3 ...
6
votes
3answers
872 views

Invert and subtract, is there any explanation?

I see in many Brazilian sites that, if you get a number and subtract it by its reverse, you will have zero or a multiple of nine. For example: ...
4
votes
2answers
718 views

The sum of digits of $3(3x+3)$ is always $9$ for any $x$ between $1$ and $9$

Given the following 'joke' I stumbled across today It's easy enough to figure out that the answer is always 9. Asshole. However when I tried to 'prove' this for ...
5
votes
5answers
477 views

Why is $9 \times 11{\dots}12 = 100{\dots}08$?

While I was working on Luhn algorithm implementation, I discovered something unusual. $$ 9 \times 2 = 18 $$ $$ 9 \times 12 = 108 $$ $$ 9 \times 112 = 1008 $$ $$ 9 \times 1112 = 10008 $$ Hope you can ...

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