Find the exponent of 3 in 100!

To find the exponent of 3 in the factorial of 100 (100!), we can use a method that involves dividing 100 by powers of 3 and summing the integer results. This helps us determine how many times 3 is a factor in the numbers from 1 to 100.

Steps to Calculate the Exponent

• Divide 100 by 3 and take the integer part: 100 ÷ 3 = 33
• Next, divide 100 by 3² (which is 9): 100 ÷ 9 = 11
• Then, divide 100 by 3³ (which is 27): 100 ÷ 27 = 3
• Finally, divide 100 by 3⁴ (which is 81): 100 ÷ 81 = 1

Summing the Results

Now, we add all these integer results together:

• 33 (from 3)
• 11 (from 9)
• 3 (from 27)
• 1 (from 81)

Adding these gives us:

33 + 11 + 3 + 1 = 48

The Final Answer

Thus, the exponent of 3 in 100! is 48.