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(c) By applying divisibility rules and long division, we find that 289 is divisible by 17.
Therefore, it is not a prime number. It is a composite number.

l Numbers between 100 and 200
We know that 15 × 15 = 225 > 200. So, if the given number is divisible by any prime number less
Note: than 15, it is a composite number. Otherwise, it is a prime number.
l Numbers between 100 and 400
We know that 20 × 20 = 400. So, if the given number is divisible by any prime numbers less than
20, it is a composite number. Otherwise, it is a prime number.

Prime Conjectures

There are many mathematical statements that are believed to be true based on evidence or
observations but not yet been formally proven. Such a statement is called a conjecture. We have
discussed about conjecture – Collatz Conjecture earlier.

Goldbach Conjecture
In prime numbers, a famous conjecture is Goldbach conjecture given by Christian Goldbach in
year 1742 that states:
Every number greater than 2 can be written as the sum of two or more primes.

For example, 7 = 2 + 5
29 = 5 + 11 + 13.
Example 19: Express:
(a) 42 as the sum of two odd primes. (b) 53 as the sum of three odd primes.

Solution: (a) 42 = 5 + 37 In such questions, more than one number
(b) 53 = 3 + 7 + 43 Note: combinations are possible.
Twin Prime Conjecture

There are infinitely many pairs of twin primes.

For example, (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), ....
Example 20: Can you find out how many prime numbers less than 75 will leave an odd reminder
when divided by 5?
Solution: The number will be in the form of 5 × quotient + 1 or 5 × quotient + 3
On guessing we have the following numbers that leaves an odd remainder, when divided by 5.

1, 3, 11, 13, 16, 18, 21, 23, 26, 28, 31, 33, 36, 38, 41, 43, 46, 48, 51, 53, 56, 58, 61, 63, 66, 68, 71 and 73
After removing the ones that are not prime, we are left out with primes only and they are:
3, 11, 13, 23, 31, 41, 43, 53, 61, 71 and 73.

Prime Art

You can place prime numbers on a circle and get a beautiful design. Place counting numbers on a
circle in such a way that the sum and differences of every pair of adjacent numbers is prime. Let’s
place numbers 1 to 14 on a circle such that the sum and differences of two consecutive numbers
are prime.

157 Prime Time