Determining the First Non-Zero Digit of 50! Without a Calculator: A Step-by-Step Approach
The first non-zero digit of 50! is a fascinating mathematical puzzle. This article walks you through the method to find this digit without the aid of a calculator. Let's dive into the process step by step.
Introduction to Factorials and Trailing Zeros
A factorial, such as 50!, is the product of all positive integers up to 50. However, to determine the first non-zero digit, it's essential to understand the concept of trailing zeros, which are zeroes at the end of the number. Trailing zeros in a factorial are caused by the factors of 10, which are themselves composed of pairs of the factors 2 and 5. Since there are generally more factors of 2 than 5, we only need to count how many times 5 is a factor in 50!.
Step 1: Calculating Trailing Zeros
Let's calculate the number of trailing zeros in 50! using the formula:
Number of trailing zeros ?n/5? ?n/25? ?n/125? [...]
For n 50:
Floor(50/5) 10
Floor(50/25) 2
Therefore, 50! has 12 trailing zeros.
Step 2: Removing Trailing Zeros
To find the first non-zero digit, we remove the trailing zeros by dividing 50! by 10^12. This step simplifies the factorial without affecting its non-zero digits.
Step 3: Considering Remaining Factors
After removing the factors of 10, we need to consider the product of the remaining factors. This involves calculating the product of all numbers from 1 to 50, excluding the factors of 2 and 5.
Step 3.1: Counting Factors of 2 and 5
To count the factors of 2 in 50!:
Floor(50/2) Floor(50/4) Floor(50/8) Floor(50/16) Floor(50/32) 25 12 6 3 1 47
And we already calculated that there are 12 factors of 5.
Step 3.2: Net Factors of 2
The net number of factors of 2 is:
47 - 12 35
Step 4: Calculating the Product Modulo 10
Next, we calculate the product of the numbers from 1 to 50, excluding the factors of 2 and 5, modulo 10. This involves considering the product of the odd numbers and the remaining even numbers.
The product of odd numbers from 1 to 49 modulo 10:
1 × 3 × 5 × 7 × 9 × ... × 49
The product modulo 10 can be simplified as:
1 × 3 × 5 × 7 × 9 × 11 × 13 × 15 × 17 × 19 × 21 × 23 × 25 × 27 × 29 × 31 × 33 × 35 × 37 × 39 × 41 × 43 × 45 × 47 × 49 ≡ 5 (mod 10)
Step 5: Remaining Factors of 2 Modulo 10
The remaining factors of 2 contribute to the product modulo 10. We need to calculate:
2^35 mod 10
The powers of 2 modulo 10 cycle every 4:
2^1 ≡ 2, 2^2 ≡ 4, 2^3 ≡ 8, 2^4 ≡ 6
Therefore, 35 mod 4 3, so 2^35 ≡ 8 (mod 10)
The net product modulo 10 is then:
5 × 8 40 ≡ 0 (mod 10)
Since we are looking for the first non-zero digit, we need to consider the product of these factors without modulo 10:
5 × 8 40
Final Step: Combining Results
Combining the above results, the first non-zero digit of 50! is:
2
Conclusion
The first non-zero digit of 50! is 2. This method demonstrates a systematic approach to finding the first non-zero digit of a factorial without using a calculator, emphasizing the importance of understanding the properties of factorials and modular arithmetic.