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To test the divisibility of a number by a power of 2 or a power of 5 (2 n or 5 n, in which n is a positive integer), one only need to look at the last n digits of that number. To test divisibility by any number expressed as the product of prime factors p 1 n p 2 m p 3 q {\displaystyle p_{1}^{n}p_{2}^{m}p_{3}^{q}} , we can separately test for ...
Two properties of 1001 are the basis of a divisibility test for 7, 11 and 13. The method is along the same lines as the divisibility rule for 11 using the property 10 ≡ -1 (mod 11). The two properties of 1001 are 1001 = 7 × 11 × 13 in prime factors 10 3 ≡ -1 (mod 1001) The method simultaneously tests for divisibility by any of the factors ...
A primality test is an algorithm for determining whether an input number is prime. ... since divisibility by an even number implies divisibility by 2. ...
This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime. [2] Once all the multiples of each discovered prime have been marked as composites, the remaining unmarked numbers are primes.
If p is composite, the probability that the test declares it a strong probable prime anyway is at most 1 ⁄ 4, in which case p is a strong pseudoprime, and a is a strong liar. Therefore after k non-conclusive random tests, the probability that p is composite is at most 4 −k, and may thus be made as low as desired by increasing k.
Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10.
Sanity tests may sometimes be used interchangeably with smoke tests [3] insofar as both terms denote tests which determine whether it is possible and reasonable to continue testing further. On the other hand, a distinction is sometimes made that a smoke test is a non-exhaustive test that ascertains whether the most crucial functions of a ...
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.