Divisibility of 9 trick: why does it work? [duplicate] I've been wondering ever since I learned about the divisibility of nine trick Add up the sum of the digits If the sum is a multiple of nine, the entire number is divisible by nine If the sum is too big to tell, repeat with the sum It's amazing But my question is: why does this work? Does it have something to do with the fact that 10 - 9 = 1?
Is $b\\mid a$ standard notation for $b$ divides $a$? This is the standard way, in the specific meaning of compliance to international standards: ISO 80000-2, clause 2 7-17 Note that the vertical bar character used there is normatively identified as U+2223 DIVIDES (∣), note the common U+007C VERTICAL LINE (|) that we enter directly on a keyboard It is of course possible to express the same thing using a congruence notation, but only for
How to prove the divisibility rule for $3\, $ [casting out threes] iff xyz= (100x)+ (10y)+z=x+y+z=0 (mod3) Therefore x+y+z=0 (mod 3), meaning that the sum of the digits is divisible by 3 This is an if and only if statement You can generalize it to n digit numbers The idea is to express the n digit numbers in powers of 10 Since powers of 10=1 (mod 3), the digit is divisible by 3 iff the sum is divisible by 3
Proof for divisibility by $7$ - Mathematics Stack Exchange There is also a similar less known trick for divisibility by 11 Since $10 = -1 \mod 11$, if you add the digits of a number in reverse order, alternating signs, and get something that is divisible by 11, then your original number is divisible by 11 For example for $1617$ you have $7-1+6-1 = 11$ so $1617$ is divisible by $11$
Divisibility by 7 - Mathematics Stack Exchange The best way to test if a number is divisible by any other number is by deducting the number n times and check whether the remainder is divisible by n for example 861-7n 7
elementary number theory - Divisibility Tests for Palindromes . . . A palindrome is divisible by 27 if and only if its digit sum is A palindrome is divisible by 81 if and only if its digit sum is This doesn't straightforwardly extend to 243 As an example, using this you can immediately see the smallest palindromic multiple of 81 is 999999999, and the smallest palindromic multiple of 405 is 54999999945, etc