Divisibility rule

Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American.

Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor.

To test divisibility by any number expressed as the product of prime factors

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).

Also, one can simply divide the number by 2, and then check the result to find if it is divisible by 2.

For example, the number 40 ends in a zero, so take the remaining digits (4) and multiply that by two (4 × 2 = 8).

For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25).

If it is divisible by 2 continue by adding the digits of the original number and checking if that sum is a multiple of 3.

All it would take with this simplification is to memorize the sequence above (132645...), and to add and subtract, but always working with one-digit numbers.

Note: The reason why this works is that if we have: a+b=c and b is a multiple of any given number n, then a and c will necessarily produce the same remainder when divided by n. In other words, in 2 + 7 = 9, 7 is divisible by 7.

Repeat that method of multiplying the units digit by five and adding that product to the number of tens.

This method could be useful in a mathematics competition such as MATHCOUNTS, where time is a factor to determine the solution without a calculator in the Sprint Round.

For example: Because 1001 is divisible by seven, an interesting pattern develops for repeating sets of 1, 2, or 3 digits that form 6-digit numbers (leading zeros are allowed) in that all such numbers are divisible by seven.

Notice that leading zeros are permitted to form a 6-digit pattern.

This phenomenon forms the basis for Steps B and C. Step B: If the integer is between 1001 and one million, find a repeating pattern of 1, 2, or 3 digits that forms a 6-digit number that is close to the integer (leading zeros are allowed and can help you visualize the pattern).

Then, break the integer into a smaller number that can be solved using Step B.

For example: This allows adding and subtracting alternating sets of three digits to determine divisibility by seven.

Understanding these patterns allows you to quickly calculate divisibility of seven as seen in the following examples: Pohlman–Mass method of divisibility by 7, examples: Multiplication by 3 method of divisibility by 7, examples: Finding remainder of a number when divided by 7 7 − (1, 3, 2, −1, −3, −2, cycle repeats for the next six digits) Period: 6 digits.

The correctness of the method is then established by the following chain of equalities: Let N be the given number

Method In order to check divisibility by 11, consider the alternating sum of the digits.

Using the second sequence, Answer: 7 × 1 + 6 × 10 + 5 × 9 + 4 × 12 + 3 × 3 + 2 × 4 + 1 × 1 = 178 mod 13 = 9 Remainder = 9 A recursive method can be derived using the fact that

Divisibility properties of numbers can be determined in two ways, depending on the type of the divisor.

For instance, one cannot make a rule for 14 that involves multiplying the equation by 7.

This is not an issue for prime divisors because they have no smaller factors.

Likewise, since 10 × (28) = 280 = 1 mod 31 also, we obtain a complementary rule y + 28x of the same kind - our choice of addition or subtraction being dictated by arithmetic convenience of the smaller value.

Then add 1 and divide by 10, denoting the result as m. Then a number N = 10t + q is divisible by D if and only if mq + t is divisible by D. If the number is too large, you can also break it down into several strings with e digits each, satisfying either 10e = 1 or 10e = −1 (mod D).

The piece wise form of D(n) and the sequence generated by it were first published by Bulgarian mathematician Ivan Stoykov in March 2020.

Many of the simpler rules can be produced using only algebraic manipulation, creating binomials and rearranging them.

This is because sufficiently high powers of the base are multiples of the divisor, and can be eliminated.

This section will illustrate the basic method; all the rules can be derived following the same procedure.