Question

The number of zeros at the end of \(100 !\) is (a) 20 (b) 22 (c) 24 (d) 26

Short answer

The number of zeros at the end of \(100!\) is determined by the number of 5s in its prime factors. Using the formula \[f_p(n) = \frac{n - s_p(n)}{p - 1}\], we can find the number of 5s in \(100!\) as 24. Therefore, there are 24 zeros at the end of \(100!\). Answer: \((c) 24\).

Step-by-step solution

Write the formula to find the number of occurrences of a prime factor in the factorial

To find the number of occurrences of a prime factor 'p' in the factorial, we can use the formula \[f_p(n) = \frac{n - s_p(n)}{p - 1}\] where \(s_p(n)\) is the sum of the digits of \(n\) when expressed in base p and \(f_p(n)\) is the number of times a prime factor p occurs in the factorial n.

Find the number of 5s in 100!

We have to find the number of 5s in 100! using the formula from Step 1. Here, \(n = 100\) and \(p = 5\). First, let's find \(s_p(n) = s_5(100)\), which is the sum of the digits of 100 when expressed in base 5. \[100_{10} = 400_{5}\] The sum of the digits of 100 in base 5 is 4. Now we can find the number of occurrences of 5 in 100! as: \[f_5(100) = \frac{100 - 4}{5 - 1} = \frac{96}{4} = 24\]

Determine the number of zeros at the end of 100!

The number of zeros at the end of 100! is equal to the number of 5s in the 100!, which we found in Step 2. Therefore, there are 24 zeros at the end of 100!. The correct option is: (c) 24

Key concepts

Prime Factorization

Understanding prime factorization is crucial when dealing with factorials, especially in problems involving trailing zeros. Prime factorization involves breaking down a number into its prime components, which are numbers that can only be divided by themselves and 1 without leaving a remainder. When calculating the number of zeros at the end of a factorial, we focus on the prime factor 5. This is because every factor of 10 in the factorial contributes a zero, and each factor of 10 is made of a pair of 2 and 5. Since factorials typically contain more 2s than 5s, the number of 5s determines the number of zeros.To find the power of the prime factor in a factorial, we use the formula \[ f_p(n) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \]where \(p\) is the prime factor and \(n\) is the number whose factorial we are evaluating. This formula considers how many times a power of the prime number divides the factorial, giving us the total count of that prime.

Base Conversion

Base conversion is the process of changing a number from one base to another. Each position in a number expressed in a particular base has a certain worth, which depends on the base.In the context of our exercise, converting a number to base 5 was important for calculating the sum of its digits. When we converted 100 from base 10 to base 5, we noticed that \[ 100_{10} = 400_{5} \] This conversion is carried out by dividing the number repeatedly by 5 and recording the remainders. Each remainder represents a digit in the new base.Knowing how to convert between bases is essential in mathematics and computer science, where different numbering systems are frequently used. By understanding base conversion, you can tackle different numerical problems and understand their structures in various systems.

Digit Sum in Bases

The sum of digits in a particular base is essential in determining certain properties of numbers, such as in the formula for the occurrences of a factor in factorials. In our step-by-step solution, we calculated the sum of the digits of 100 in base 5.After converting 100 to base 5, we obtained "400". The sum of these digits is \(4 + 0 + 0 = 4\).This calculation was instrumental when using the formula for the prime factors in factorials:\[ f_p(n) = \frac{n - s_p(n)}{p - 1} \]Here, \(s_p(n)\) represents the sum of digits in base \(p\), which helps adjust the estimation for how many times a prime factor appears in the factorial. By including the digit sum, we account for any excess that might not fully contribute to the factor of the number we're analyzing.