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To test divisibility by any number expressed as the product of prime factors , we can separately test for divisibility by each prime to its appropriate power. For example, testing divisibility by 24 (24 = 8 × 3 = 2 3 × 3) is equivalent to testing divisibility by 8 (2 3 ) and 3 simultaneously, thus we need only show divisibility by 8 and by 3 ...
In arithmetic, for example, when multiplying by 9, using the divisibility rule for 9 to verify that the sum of digits of the result is divisible by 9 is a sanity test—it will not catch every multiplication error, but is a quick and simple method to discover many possible errors.
Given an integer n (n refers to "the integer to be factored"), the trial division consists of systematically testing whether n is divisible by any smaller number. Clearly, it is only worthwhile to test candidate factors less than n, and in order from two upwards because an arbitrary n is more likely to be divisible by two than by three, and so on.
The first deterministic primality test significantly faster than the naive methods was the cyclotomy test; its runtime can be proven to be O((log n) c log log log n), where n is the number to test for primality and c is a constant independent of n. Many further improvements were made, but none could be proven to have polynomial running time.
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
Rule of nines or rule of nine may refer to: . Rule of nine (linguistics), an orthographic rule of the Ukrainian language. Rule of nines (mathematics), a test for divisibility by 9 involving summing the decimal digits of a number
The first step is to test n for divisibility by each of the members of P; clearly if n is divisible by one of these primes, then we are finished already. However, 187 is not divisible by 2, 3, 5, or 7. Next, we search for suitable values of z; the first few are 2, 5, 9, and 56.
Digit sums and digital roots can be used for quick divisibility tests: a natural number is divisible by 3 or 9 if and only if its digit sum (or digital root) is divisible by 3 or 9, respectively. For divisibility by 9, this test is called the rule of nines and is the basis of the casting out nines technique for checking calculations.