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m is a divisor of n (also called m divides n, or n is divisible by m) if all prime factors of m have at least the same multiplicity in n. The divisors of n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then ...
If none of its prime factors are repeated, it is called squarefree. (All prime numbers and 1 are squarefree.) For example, 72 = 2 3 × 3 2, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Euler diagram of numbers under 100:
Many 19th century mathematicians still considered to be prime, [42] and Derrick Norman Lehmer included in his list of primes less than ten million published in 1914. [43] Lists of primes that included 1 continued to be published as recently as 1956. [44] [45]
The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which n/m is again an integer (which is necessarily also a divisor of n). For example, 3 is a divisor of 21, since 21/7 = 3 (and therefore 7 is also a divisor of 21). If m is a divisor of n, then so is −m. The tables below only ...
Not all e i ≡ 2 (mod 5). [42] If all e i ≡ 1 (mod 3) or 2 (mod 5), then the smallest prime factor of N must lie between 10 8 and 10 1000. [42] More generally, if all 2e i +1 have a prime factor in a given finite set S, then the smallest prime factor of N must be smaller than an effectively computable constant depending only on S. [42]
The same method can also be illustrated with a Venn diagram as follows, with the prime factorization of each of the two numbers demonstrated in each circle and all factors they share in common in the intersection. The lcm then can be found by multiplying all of the prime numbers in the diagram. Here is an example: 48 = 2 × 2 × 2 × 2 × 3,
45 is an odd number and a Størmer number, as well as the 9th triangular number [1] and 5th hexagonal number. [ 2 ] 45 degrees is half of a right angle. It is also the smallest positive number that can be expressed as the difference of two nonzero squares in more than two ways: 7 2 − 2 2 {\displaystyle 7^{2}-2^{2}} , 9 2 − 6 2 ...
It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42. ... 21, 42. The set of all positive divisors of 60, = ...