#### To determine

**To find:**

We have to find the total number of primes less than or equal to 25 and 100 i.e. we have to find

π25,π(100).

#### Answer

π25=9, π100=25

#### Explanation

**Calculation:**

To find π25, we have to find all prime numbers less than or equal to 25.

Prime numbers less than 25 are 2,3,5,7,11,13,17,19,23.

So, π25=9.

Now, we find prime numbers less than or equal to 100.

Prime numbers less than 100 are

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,57,59,61,67,83,87,89,91,97.

So, π100=25.

**Conclusion:**

π25=9, π100=25

#### To determine

**To show:** We have to prove the prime number theorem by computing the ratio of π(n) to nlnn for n=100,1000,10,000,100,000,1,000,000,10,000,000

#### Answer

**Answer:nlnn**.

#### Explanation

**Calculation:**

n |
π(n) |
nlnn |
π(n)nlnn |

100 |
25 |
22 |
1.15 |

1,000 |
168 |
145 |
1.16 |

10,000 |
1,229 |
1,086 |
1.13 |

100,000 |
9,592 |
8,686 |
1.10 |

1,000,000 |
78,498 |
72,382 |
1.08 |

10,000,000 |
664,579 |
620,420 |
1.07 |

We can see that the ratio π(n)nlnn is almost equal to 1. Hence, we can estimate π(n) by nlnn.

**Conclusion:nlnn**.

#### To determine

**To find:**

We have to estimate the number of primes less than to a billion using the prime number theorem.

#### Answer

**Answer: 48254942**

#### Explanation

**Calculation:**

By prime number theorem we can estimate the total number of primes less than equal to n by computing nlnn.

So

π1,000,000,000≈1,000,000,000ln1,000,000,000≈48254942

**Conclusion: 48254942**