Number Theory

• Prime numbers are numbers having exactly two factors: 1 and the number itself. For example, the factors of 23 are 1 and 23 only. So, 23 is a prime number.
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• Composite numbers are numbers having more than two factors. For example, the factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. Since there are 8 factors of 42, it is a composite number.
• 1 is neither prime nor composite as it has exactly one factor.
• The smallest prime number is 2.
• The smallest even prime number is 2 and the smallest odd prime number is 3.
• All even numbers except 2 are composite.
• The pairs of prime numbers whose difference is 2 are known as twin primes. For example, 11 and 13 are twin primes.

• According to fundamental theorem of arithmetic, a number can be represented as the product of primes having a unique factorisation.

Example: 

Check whether 15ⁿ in divisible by 10 or not for any natural number n. Justify your answer.

Solution:

A number is divisible by 10 if it is divisible by both 2 and 5.

15ⁿ = (3 . 5)ⁿ

3 and 5 are the only primes that occur in the factorisation of 15ⁿ

By uniqueness of fundamental theorem of arithmetic, there is no o…