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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.
  • 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.


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


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…

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