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Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are in bold and superior highly composite numbers are starred. ... 24, 33, 36, 44 ...
The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4; [2] [3] this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any number ends in a two digit number that ...
Since it is possible to find sequences of 36 consecutive integers such that each inner member shares a factor with either the first or the last member, 36 is an Erdős–Woods number. [11] The sum of the integers from 1 to 36 is 666 (see number of the beast). 36 is also a Tridecagonal number. [12]
33 2 = 1089 34 2 = 1156 35 2 = 1225 36 2 = 1296 ... up with itself to yield the square number, while other divisors come in pairs. ... number is divisible both by 2 ...
Every third odd number is divisible by 3, and therefore no three successive odd numbers can be prime unless one of them is 3. Therefore, 5 is the only prime that is part of two twin prime pairs. The lower member of a pair is by definition a Chen prime. If m − 4 or m + 6 is also prime then the three primes are called a prime triplet.
A number that belongs to a singleton club, because no other number is friendly with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, if the numbers n and σ( n ) are coprime – meaning that the greatest common divisor of these numbers is 1, so that σ( n )/ n is an ...
Informally, the probability that any number is divisible by a prime (or in fact any integer) p is ; for example, every 7th integer is divisible by 7. Hence the probability that two numbers are both divisible by p is 1 p 2 , {\displaystyle {\tfrac {1}{p^{2}}},} and the probability that at least one of them is not is 1 − 1 p ...
The prime numbers are precisely the atoms of the division lattice, namely those natural numbers divisible only by themselves and 1. [ 2 ] For any square-free number n , its divisors form a Boolean algebra that is a sublattice of the division lattice.