Let n be an integer with last digit d.

Repeat until divisibility of n by 7 has been decided:

- Set m to n with last digit removed.

- Set n to m - 2d

- Determine if n is divisible by 7.

Proof:

m = (n - d)/10

m = (n - d)/10 - 2d

m = ( n - 21d ) / 10

If 7 / n then 7 / ( n - 21d ) / 10 since (7,10)=1 and 7/21.

Example:

n = 8 641 969

m = 864 196

n = 864 196 - 18 = 864 178

m = 86 417

n = 86 417- 16 = 86 401

m = 8 640

n = 8640 - 2 = 8638

m = 863

n = 863 - 16 = 847

m = 84

n = 84 - 14 = 70

Divisible by 7.

I suppose trivia like this are only interesting for math(s)(*) enthusiasts and number geeks. Computational number theorists working with numbers of several million digits may actually use methods like this.

P.S.

(*) Just learned from Math is Fun that "Mathematics is commonly called Math in the US and Maths in the UK and in many other countries.".

2-2018 Teaching by misleading

2 months ago

To know the first real rule for divisibility by 7 of the History of Number Theory watch this video:

ReplyDeletehttp://www.youtube.com/watch?v=ZUozMuPE1RA