# Linked Questions

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### 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 $x = 4518.$ ...
3answers
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### 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 ...
1answer
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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 $$\... 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 ... 1answer 29 views ### Prove  n \equiv s(n)\ (mod\ 3) using the fact that \ [10^n] = . [duplicate] Prove  n \equiv s(n)\ (mod\ 3) using the fact that \ [10^n] = . 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}+ \... 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. ... 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 = (... 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! 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 ... 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 ... 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 ... 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 ... 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: ... 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 ... 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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