if the highest power of 225 in n! is 10, find the number of values that n can assume
Answers
Answer:
The number of values that n can assume is 2.
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 = maximum 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.
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Answer:
The correct answer is : greater than or equal to 82
Step-by-step explanation:
We know that 225 = 3^2 x 5^2, and we need to find the largest value of n such that the highest power of 225 in n! is 10.
To find this value, we need to consider how many times the prime factors 3 and 5 appear in n!. Since 225 = 3^2 x 5^2, we need to find the largest value of n such that n! has at least 10 factors of 3 and 10 factors of 5.
We know that the number of factors of a prime p in n! is given by the expression: (n/p) + (n/p^2) + (n/p^3) + ...
So we need to find the smallest value of n such that:
(n/3) + (n/9) + (n/27) + ... >= 10, and
(n/5) + (n/25) + (n/125) + ... >= 10
We can solve these inequalities using trial and error, or we can use a calculator or computer program to find that n = 82 is the smallest value that satisfies both inequalities.
Therefore, the number of possible values of n is infinite, since any value of n greater than or equal to 82 will also have a highest power of 225 in n! equal to 10.
To learn more about inequalities, visit:
https://brainly.in/question/12743729
To learn more about factors, visit:
https://brainly.in/question/17688561
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