Divisibility July 2026

THEN divisibility by B-1 in base B can be determined by adding the digits.

This is not an original idea, turns out it’s well known.

Divisibility by 3 or 9 (from the wiki)
First, take any number (for this example it will be 492) and add together each digit in the number (4 + 9 + 2 = 15). Then take that sum (15) and determine if it is divisible by 3. The original number is divisible by 3 (or 9) if and only if the sum of its digits is divisible by 3 (or 9).

Adding the digits of a number up, and then repeating the process with the result until only one digit remains, will give the remainder of the original number if it were divided by nine (unless that single digit is nine itself, in which case the number is divisible by nine and the remainder is zero).

This can be generalized to any standard positional system, in which the divisor in question then becomes one less than the radix; thus, in base-twelve, the digits will add up to the remainder of the original number if divided by eleven, and numbers are divisible by eleven only if the digit sum is divisible by eleven.

THEN divisibility by B+1 in base B can be determined by the same process.

This is not an original idea, either.

Add the digits of 64₈: 6₈ + 4₈ = 12₈

Add the digits again: 1₈ + 2₈ = 3₈, therefore a remainder of 3₈

52₁₀ = 7₁₀ x 7₁₀ remainder 3₁₀.

64₈ = 7₈ x 7₈ remainder 3₈.

Yes, 3₈ = 3₁₀

Add the even digits, and subtract the odd digits, of 137₈: 1₈ – 3₈ + 7₈ = 5₈

95₁₀ = 9₁₀ x 10₁₀ remainder 5₁₀.

137₈ = 11₈ x 12₈ remainder 5₈.

Yes, 5₈ = 5₁₀