500! is divisible by 21. How many times can 21 go into 500! ?
The answer key does not simply state 500!/21 times, and my calculator does not have the capacity to compute such a large sum. Thanks!
do you mean 5001 ???
5001 divided by 21 ?
go to google and type in wolframalpha and on the website type in 5001 divided by 21 ? if that's your question.
It would go 500!/21 times
This is a very "large" number , it would be appr 500 digits long.
Considering that 1 Billion has only 10 digits, you can imagine the enormity of this number.
What are the choices?
To find out how many times 21 can divide into 500!, we need to consider the prime factorization of both 21 and 500!.
First, let's factorize 21 into its prime factors: 21 = 3 * 7.
Now, let's consider the prime factorization of 500!: 500! = 2^a * 3^b * 5^c * ... (where 'a', 'b', 'c', and so on are the exponents).
To determine how many times 21 can divide into 500!, we need to find the minimum power of each prime factor in the prime factorization of 500!. Since we have a prime factorization of 3 and 7 for 21, we need to determine the power of 3 and 7 in the prime factorization of 500!.
We can use the formula to find the power of a prime in the prime factorization of N!, where N is any positive integer:
Power of a prime 'p' in N! = (N / p) + (N / p^2) + (N / p^3) + ... + (N / p^k)
(Note: In the above formula, p^k is the largest power of p that does not exceed N.)
Using this formula, let's calculate the power of 3 and 7 in 500!:
Power of 3 in 500! = (500 / 3) + (500 / 3^2) + (500 / 3^3) + ... + (500 / 3^k)
(In this case, k will be the largest power of 3 that does not exceed 500.)
Similarly,
Power of 7 in 500! = (500 / 7) + (500 / 7^2) + (500 / 7^3) + ... + (500 / 7^k)
(Again, k will be the largest power of 7 that does not exceed 500.)
By calculating these expressions, we can determine the power of 3 and 7 in 500!. Then, we take the smaller of the powers as the answer, as the larger power will not contribute to the divisibility by 21.
Since the powers of 3 and 7 in 500! are both finite and computable, you can perform these calculations by writing a program or by using a software tool like a programming language or a symbolic math system.