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a prime number has only 1 and itself as divisors; that is, d(n) = 2; a composite number has more than just 1 and itself as divisors; that is, d(n) > 2; a highly composite number has a number of positive divisors that is greater than any lesser number; that is, d(n) > d(m) for every positive integer m < n.
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N. For example, 6 is highly composite because d(6)=4 and d(n)=1,2,2,3,2 for n=1,2,3,4,5 respectively.
An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital. gcd( m , n ) ( greatest common divisor of m and n ) is the product of all prime factors which are both in m and n (with the smallest multiplicity for m and n ).
For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and () is even; for a square integer, one divisor (namely ) is not paired with a distinct divisor and () is odd. Similarly, the number σ 1 ( n ) {\displaystyle \sigma _{1}(n)} is odd if and only if n is a square or twice a square.
More generally, an a-by-b rectangle can be covered with square tiles of side length c only if c is a common divisor of a and b. For example, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares.
A positive divisor of that is different from is called a proper divisor or an aliquot part of (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide n {\displaystyle n} but leaves a remainder is sometimes called an aliquant part of n . {\displaystyle n.}
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. The integer 12 is the first abundant number.
This is true in the case of 6; 6's divisors are 1,2,3, and 6, but an abundant number is defined to be one where the sum of the divisors, excluding itself, is greater than the number itself; 1+2+3=6, so this condition is not met (and 6 is instead a perfect number). However all colossally abundant numbers are also superabundant numbers. [12]