Pascal's Triangle

On Number

"Number is the beginning and end of thought. Thought gave birth to number but reaches not beyond. "

--Gösta Mittag-Leffler, 1903

Rules of Divisibility

An integer N is evenly divisible...

by 2: if it is even--this depends only on the last digit;

by 3: if the sum of its digits is divisible by 3;

by 4: if the number formed by the last two digits is divisible by 4;

by 5: if it ends in 5 or 0;

by 6: if it is divisible by 2 and 3;

by 7: if the number formed after cancellation of the units digit and subtraction of twice the value of the units digit is divisible by 7;

by 8: if the number formed by the last three digits is divisible by 8;

by 9: if the sum of its digits is divisible by 9;

by 10: if it ends in 0;

by 11: if the difference between the cross sums of alternate digits is divisible by 11 (presupposes sufficiently high N);

by 12: if it is divisible by 3 and 4.

Example 1

The number N = 4893 can be shown to be divisible by 7:

4893 ---> 489 - 2*3 = 483 = 69*7.

Continuing the criterion,

48 - 2*3 = 42 ---> 4 - 2*2 =0, which is divisible by 7.

Example 2

The number N = 6,793,522,179,658,192 can be show to be divisible by 11:

6 + 9 + 5 + 2 + 7 + 6 + 8 + 9 = 52

7 + 3 + 2 + 1 + 9 + 5 + 1 + 2 = 30

The difference of these two cross sums is 22 ---> 2*11.

For a list of divisibility rules beyond 12, see here.