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Animation showing an application of the Euclidean algorithm to find the greatest common divisor of 62 and 36, which is 2. A more efficient method is the Euclidean algorithm , a variant in which the difference of the two numbers a and b is replaced by the remainder of the Euclidean division (also called division with remainder ) of a by b .
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
The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 (sequence A001597 in the OEIS). 1 is sometimes included. A powerful number (also called squareful ) has multiplicity above 1 for all prime factors.
A number that has the same number of digits as the number of digits in its prime factorization, including exponents but excluding exponents equal to 1. A046758: Extravagant numbers: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, ... A number that has fewer digits than the number of digits in its prime factorization (including ...
This result suffices to show that the number of steps in Euclid's algorithm can never be more than five times the number of its digits (base 10). [101] For if the algorithm requires N steps, then b is greater than or equal to F N+1 which in turn is greater than or equal to φ N−1, where φ is the golden ratio.
A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2.
This representation is commonly extended to all positive integers, including 1, by the convention that the empty product is equal to 1 (the empty product corresponds to k = 0). This representation is called the canonical representation [10] of n, or the standard form [11] [12] of n. For example, 999 = 3 3 ×37, 1000 = 2 3 ×5 3, 1001 = 7×11×13.
180 is a 61-gonal number, [2] while 61 is the 18th prime number. Half a circle has 180 degrees, [7] and thus a U-turn is also referred to as a 180. Summing Euler's totient function φ(x) over the first + 24 integers gives 180. In binary it is a digitally balanced number, since its binary representation has the same number of zeros as ones ...