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.".
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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