Modular Arithmetic Proof for a Divisibility by 7 Test

The prolific, fast talking creator @PapaFlammy69 used this relatively obscure divisibility test for 7 when working through his solution to British Math Olympiad 2009 Problem 1. Many commenters had never heard of or seen this divisibility algorithm. His 69(soixante-neuf) position sexual reference is part of his effusive unabashed charm. proved the digits tests for both 3 and 9 in the "casting out 9s" sense without noting 10^n == 1(congruent) both modulo 3 and 9. Thx Kevin Olding

I bet Papa Flammy knows that the square root of 69 is 8 something.(Ate=8 Get it?!?)

Take the last digit of the number and double it.
Subtract this number formed from the other digits in the original number.
If this number is divisible by 7, so is the original number.
@ImprovedMath also taught this divisibility by seven criterion for integers without a rigorous proof I believe.
Quora's PHD Civil Engineer HsBadarinath July 2022 stated, without proof, an algorithm that begin with multiplying the last digit by 5 and then adding that product to the remaining digits. Iterate until a single digit remains(If it is equal to 7 then original number is divisible by 7. Some medical professionals claim math, as a hobby, prevents or slows the advance of brain maladies like Alzheimer's and dementia. Despite the nonsense promulgated on TikTok, old age/aging is NOT a disease! taylordonoghuee vacuously claims the hackneyed "age is just a number" January 2024. How profound and original! IQ is "just a number" also!

[rrndzg]

In the line that starts with "Multiplying ... " instead of <10*a_1 - 2a_0> it should put <a_1 - 2a_0>, right?