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The product of a whole number ‘n’ with every whole number until 1 is known as the factorial. The factorial of four, for example, is 4x3x2x2x1, which equals 24. The ‘!’ symbol is used to represent it. 4 equals 24! Fabian Stedman, a British author, coined the term “factorial” in 1677. Musicians would ring multiple tuned bells as part of a change ringing performance. Christian Kramp, a French mathematician, invented the factorial symbol in 1808. Number theory, algebra, geometry, probability, statistics, graph theory, and discrete mathematics are all based on the study of factorials.
What Is Factorial
The function that multiplies a number by every natural number below it is known as the factorial. “!” can be used to symbolise factorial. The product of the first n natural numbers is n factorial, which is written as n!.
Formula for n Factorial
formula for n factorial is : n!=n×(n−1)!
What is 0!
The value of the zero factorial, often known as the Factorial of 0, is 1, i.e., 0! = 1.
Here’s how it works:
1! =1
2! = 2×1=2
3! = 3×2×1=3×2! = 6
4! = 4×3×2×1=4×3! = 24
5! = 5×4×3×2×1=5×4! = 120
Factorial uses:
Permutations and combinations are an area where factorials are frequently used.
The order of the outcomes is not important in a combination.
Remainder Rule
The remainder of dividing 5,142,376,298 by 9 can be calculated using the long division method, but you can imagine how long that would take.
The good news is that the remainder rules work similarly to the divisibility rules.
These can help you discover the remainder fast without having to perform long division. Let’s stick to dividing by the numbers below:
2, 3, 4, 5, 8, 9, 10
The rest rules of 6 and 7 have been excluded because they will be discussed in a separate topic.
When dividing by two, remember to use the remainder rule. The residual is 0 if the number is divisible by 2, else it is 1. Because 1,456 is divisible by 2, there is no remainder.
2,390,399 is also not divisible by two, hence the remainder is one.
When dividing by three, remember to use the remainder rule. When you divide an integer by three, you get the same result.
When an integer is divided by three, the sum of its digits is used to get the residual. To find the remainder, divide the total by three.
When dividing by 4, use the Remainder Rule to create a two-digit number using the integer’s last two digits. The remainder is found by multiplying the two-digit number by 4.
When dividing by 5, the remainder rule applies.
Divide the last digit (the unit’s digit) by 5 to determine the remainder when dividing a number by 5.
When dividing by 8, use the Rule of the Remainder.
When dividing an integer by 8, make a three-digit number out of the last three digits. Then, using long division, divide the three-digit value by 8 to get the residual.
When dividing by 9, apply the remainder rule. When you divide an integer by nine, you get the same result.
When an integer is divided by 9, the sum of its digits is used to get the residual. To find the remaining, divide the total by nine.
When dividing by 10, apply the remainder rule. The last digit (the unit’s digit) of an integer divided by 10 equals the remainder. That’s it.
Wilson’s theorem
In number theory, Wilson’s theorem states that any prime p divides (p 1)! + 1, where n! is the factorial notation for 1 x2x 3x 4x … x n. 5 divides (5 – 1)! + 1 Equals 4! + 1 = 25. The conjecture was initially reported in Meditationes Algebraicae (1770; “Thoughts on Algebra ”) by the English mathematician Edward Waring, who attributed it to the English mathematician John Wilson.
Joseph-Louis Lagrange, a French mathematician, proved the theorem in 1771. The opposite of the theorem is likewise true: (n – 1)! + 1 is not divisible by n. These theorems provide a test for primes in theory, but the calculations are difficult for big numbers in practice.
Conclusion
Much of the factorial function’s mathematics was invented in the late 18th and early 19th centuries. Stirling’s estimate is a precise approximation of the factorial for large numbers, demonstrating that it grows faster than exponential growth. Legendre’s formula can be used to count the factorials’ trailing zeros by describing the exponents of the prime numbers throughout a prime factorization of the factorials. Except for negative integers, Daniel Bernoulli and Leonhard Euler superimposed the factorial function to a continuous function of complex numbers, the (offset) gamma function.
Binomial coefficients, double factorials, falling factorials, primorials, and subfactorials are only a few of the prominent functions and number sequences that are intimately related to the factorials. The factorial function is featured in scientific calculators and scientific computing software packages as an illustration of several computer programming approaches.
