if the highest power of 225 in! is 10, find the number of values that n can assume
Answers
Answer:
The number of values that n can assume is 2
Step-by-step explanation:
Assuming that the one who has asked the question or the one who is reading it is familiar with the concept. Note that calculating the highest power of a (prime) number involves greatest integer function.
Since, 225=(3^2) × (5^2)
Therefore maximum power of 225 = max power of 5 divided by 2
Therefore, the power of 5 has to be in the interval [20, 22). This possible when n is 85 or 95.
Thus the number of values that n can assume is 2.
Answer:
[n/225] + [n/50625] = 10
Step-by-step explanation:
To solve this problem, we need to use the formula for the highest power of a prime number that divides a factorial. The formula is given by:
highest power of p in n! = [n/p] + [n/p^2] + [n/p^3] + ...
where [] denotes the greatest integer function, and p is a prime number.
In this case, the prime number is 225 = 3^2 * 5^2. So we need to find the largest n for which the highest power of 225 in n! is 10. That is, we need to solve the equation:
[n/225] + [n/225^2] = 10
Since 225^2 = 50625, we can simplify this equation as:
[n/225] + [n/50625] = 10
Now we need to find the number of values that n can assume. To do this, we can use trial and error, or we can use a computer program or calculator. Using trial and error, we can see that n = 1125 is a solution to the equation. We can also see that n cannot be smaller than 1125, since the left-hand side of the equation would be less than 10. Therefore, the number of values that n can assume is 1, since there is only one solution to the equation.
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