Number of primes between 1 to 100 is ____.
25
Goal: Find how many prime numbers are from 1 to 100.
Step 1: Recall the meaning of a prime number.
A prime number has only two factors: 1 and the number itself.
For example, (2) is prime (factors: (1,2)).
Step 2: Note two quick facts.
• (1) is not prime (it has only one factor).
• (2) is the only even prime. Any other even number is divisible by (2), so it is not prime.
Step 3: Cross out the obvious non-primes.
• Remove all even numbers (>2): (4,6,8,10,\dots,100\).
• Remove multiples of (5) (>5): (10,15,20,25,\dots,100\).
Step 4: Check the remaining numbers by small primes.
To test if a number (n\le 100) is prime, it is enough to check divisibility by the primes (2,3,5,7\) (because (7^2=49\) and the next prime (11\) has (11^2=121>100\)).
• If a number is divisible by (3), it is not prime.
• If a number is divisible by (7), it is not prime.
Step 5: List the primes from 1 to 100.
(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\)
Step 6: Count them.
There are (25) numbers in the list above.
Conclusion:
The number of primes between (1) and (100) is (25\).