500! is divisible by 21. How many times can 21 go into 500! ?
Have zero clue how to do this. Thanks in advance!
500!/21 times
My calculator cannot compute that complicated kind of function. And plus, 500!/21 is not a part of the multiple choices.
To solve this problem, we need to understand the concept of factorials and divisibility rules.
Factorial of a number is the product of all positive integers from 1 to that number. For example, 5! (read as 5 factorial) is equal to 5 x 4 x 3 x 2 x 1 = 120.
To find out how many times 21 can go into 500!, we need to calculate the exponent of 21 in the prime factorization of 500!.
Here's how you can do it step by step:
1. Prime factorize 21: 21 = 3 x 7
2. Prime factorize 500!: Find the prime factorization of all the numbers from 1 to 500.
3. Count the frequency of 21 in the prime factorization of 500!. This can be done by counting how many times the prime factors 3 and 7 occur in the factorization.
4. To simplify the calculation, we can use the concept of divisibility by calculating the exponents of 3 and 7 in the prime factorization of 500! and divide them by the exponents of 3 and 7 in the prime factorization of 21.
Now let's calculate the exponents:
Exponents of 3 in 500! = ⌊500/3⌋ + ⌊500/9⌋ + ⌊500/27⌋ + ... (continue until the quotient becomes 0)
Exponents of 7 in 500! = ⌊500/7⌋ + ⌊500/49⌋ + ⌊500/343⌋ + ... (continue until the quotient becomes 0)
⌊x⌋ represents the largest integer less than or equal to x.
5. Divide the exponents obtained in step 4 by the exponents of 3 and 7 in the prime factorization of 21 (which are 1 each).
The smaller of the two quotients will give you the number of times 21 can go into 500!
So, to summarize, follow these steps:
1. Prime factorize 21.
2. Prime factorize 500!.
3. Calculate the exponents of 3 and 7 in the prime factorization of 500!.
4. Divide the exponents by 1 (the exponents of 3 and 7 in the prime factorization of 21) to get the quotients.
5. Take the smaller quotient as the answer.
Note: The calculation of factorials and prime factorizations for large numbers can be time-consuming. You may need to use software, online calculators, or programming tools to perform these calculations efficiently.