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Example 2: The number to be tested is 15751537186. (1 × 15 − 3 × 75 + 2 × 15) + (1 × 37 − 3 × 18 + 2 × 60) = −180 + 103 = −77 The result −77 is divisible by seven, thus the original number 15751537186 is divisible by seven. Another digit pair method of divisibility by 7. Method
The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for certain models or applications. Infinitely divisible distributions play ...
For example 111111111111111 (15 digits) is divisible by 111 and 11111 in that base. If a number m can be expressed as a string of prime length to some base, such a number may or may not be prime, but commonly is not; for example, to base 10, there are only three such numbers of length less than 100 (1 is by definition, not prime). The three are:
For example, since 19 divides a 11 and 2 × 19 = 38 does not divide 11, so 19 divides a 18k+11 for all k. Thus, the definition of irregular pair ( p , n ) , n should be at most p − 2 . The following table shows all irregular pairs with odd prime p ≤ 661 :
Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter , space , time , money , or abstract mathematical objects such as the continuum .
A primality test is an algorithm for determining whether an input number is prime.Among other fields of mathematics, it is used for cryptography.Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not.
In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if > > are coprime integers, then for any integer , there is a prime number p (called a primitive prime divisor) that divides and does not divide for any positive integer <, with the following exceptions:
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.