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12: It is divisible by 3 and by 4. [6] 324: it is divisible by 3 and by 4. Subtract the last digit from twice the rest. The result must be divisible by 12. 324: 32 × 2 − 4 = 60 = 5 × 12. 13: Form the alternating sum of blocks of three from right to left. The result must be divisible by 13. [7] 2,911,272: 272 − 911 + 2 = −637.
d() is the number of positive divisors of n, including 1 and n itself; σ() is the sum of the positive divisors of n, including 1 and n itselfs() is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n
12 (twelve) is the natural number following 11 and preceding 13.. Twelve is the 3rd superior highly composite number, [1] the 3rd colossally abundant number, [2] the 5th highly composite number, and is divisible by the numbers from 1 to 4, and 6, a large number of divisors comparatively.
Demonstration, with Cuisenaire rods, of the first four highly composite numbers: 1, 2, 4, 6. A highly composite number is a positive integer that has more divisors than all smaller positive integers.
The smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5, 7, 11, 13, 17, 19, 23, and 29 (sequence A047802 in the OEIS). An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. [1]
Year 100 was a leap year starting on Wednesday of the Julian calendar. The denomination 100 for this year has been used since the early medieval period.
The number 18 is a harshad number in base 10, because the sum of the digits 1 and 8 is 9, and 18 is divisible by 9.; The Hardy–Ramanujan number (1729) is a harshad number in base 10, since it is divisible by 19, the sum of its digits (1729 = 19 × 91).
The 2-order of n is written ν 2 (n) or ord 2 (n). It is not to be confused with the multiplicative order modulo 2. The 2-order provides a unified description of various classes of integers defined by evenness: Odd numbers are those with ν 2 (n) = 0, i.e., integers of the form 2m + 1. Even numbers are those with ν 2 (n) > 0, i.e., integers of ...