Thread by @pgeerkens

@GWOMaths @MathsTim Hint in next tweet.

@GWOMaths @MathsTim There are many well known "division rules" that enable one to test the divisibility of a number by a small integer much more efficiently than performing the division.

The best known of these is Casting Out Nines:

N is divisible by 9 if and only if the sum of its digits is.

@GWOMaths @MathsTim In most, perhaps all, of these divisible rules the remainder is provided also. This is true for Casting Out Nines, resulting in a Casting Out Threes corollary - more useful to us as 3 is one of the low primes to be tested today.

@GWOMaths @MathsTim Divisible by 2: False, as odd
Divisible by 3: False as sum of digits = 2
Divisible by 5: False as units digit neither 0 nor 5

There are many Test-7 divisibility rules for 7, I believe this one is best suited to today's problem

@GWOMaths @MathsTim Divisible by 7: False as
(1*1 + 0*3 + 0*2 - ...) + ... + (... - 0*1 - 0*3 - 1*2)
= 1 + 0 + ... + 0 + (-2)
= 1 - 2
= -1 is not divisible by 7
from noting that 10^01 + 1 has 102 digits and that 102 = 6 * 17 where our set of multipliers has 6 elements.

@GWOMaths @MathsTim Casting Out Elevens

Note that Casting Out Nines is a fundamental property of nine being one less than the base of our common number system: 10. The same process works for all bases, in "casting out" the value one less than the base.

1/n

@GWOMaths @MathsTim Note also that our common number system can be thought of as a base 100 system by grouping digits in pairs, right to left.

Thus we can "Cast Out 99's" just as we Cast Out Nines;
AND
we can Cast Out Elevens just as we Cast Out Threes because 11 is a factor of 99.

2/n

@GWOMaths @MathsTim Divisible by 11: True, as:

Grouping the digits of our number in pairs, right to left, and adding we get:

01 + 00 + 00 + ... + 00 + 00 + 10 = 11 which is divisible by 11.

So our lowest prime divisor is 11.

@GWOMaths @MathsTim @threadreaderapp unroll

@GWOMaths @MathsTim Here is a comprehensive of well known divisibility rules (for small integers) from Wikipedia

en.wikipedia.org/wiki/Divisibil…

@GWOMaths @MathsTim The sequence of digits for our Rule for Seven can be remembered as:
1 = 10^0 mod 7
3 = 10^1 mod 7
2 = 10^2 mod 7
-1 = 6 mod 7 = 10^3 mod 7
-3 = 4 mod 7 = 10^4 mod 7
-2 = 5 mod 7 = 10^5 mod 7

Both the mixed sign <1,3,2,-1,-3,-2> and +ve sign <1,3,2,6,4,5> sequences work fine.