Give big-O estimates for the factorial function and the logarithm of the factorial function, where the factorial function f(n) = n! is defined by n! = 1 · 2·3· ... ·n whenever n is a positive integer, and O! = 1 . For example, 1 ! = 1, 2! = 1 · 2=2, 3 ! =1· 2·3 = 6, 4! = 1 · 2·3· 4=24. Note that the function n! grows rapidly. For instance, 20! = 2,432,902,008,176,640,000.

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Problems on numbers

Number theory is one of the most appealing topics in mathematics. Given one or more numeric values, a numeric algorithm performs some computation. The use of number theory in computing is important for applications like in the case of cryptographic algorithms. Number theory is also used in various applications such as calculating hash functions or in information security. Euclid’s algorithm is one of the oldest number-theoretic algorithms which is used to calculate the greatest common divisor of two numbers.