Question

question 22
7 parts of 1 point

a. in which of the following intervals are there more primes? why?
i. 0-99
ii. 100-199
b. what is the longest string of consecutive composite numbers in the given intervals?
c. how many twin primes are there in each interval?
d. what patterns, if any, do you see for any of the preceding questions? predict what might happen in other intervals.

part 1 of 7
a. there are primes in the interval 0-99 and primes in the interval 100-199.

part 2 of 7
there are more primes in the interval select

part 3 of 7
b. the longest string of consecutive composite numbers in the interval 0-99 is

Explanation:

Count primes in 0-99

We count prime numbers in the interval \( [0, 99] \).
The primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
There are 25 primes.

Count primes in 100-199

We count prime numbers in the interval \( [100, 199] \).
The primes are: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199.
There are 21 primes.

Compare prime density

We compare the counts from the two intervals.
Since \( 25 > 21 \), there are more primes in \( [0, 99] \).

Find consecutive composites in 0-99

We find the longest sequence of composite numbers in \( [0, 99] \).
The largest gap between consecutive primes in this range is between 89 and 97.
The composite numbers between them are: 90, 91, 92, 93, 94, 95, 96.
This sequence contains 7 consecutive composite numbers.

Answer:

Question 1

There are <blank>25</blank> primes in the interval 0-99 and <blank>21</blank> primes in the interval 100-199.

Question 2

There are more primes in the interval <blank>0-99</blank>.

Question 3

The longest string of consecutive composite numbers in the interval 0-99 is <blank>7</blank>.