Real Numbers

Co Prime Numbers

A pair of numbers of which the common factor is only one.

  1. Two prime numbers will always form a pair of co-prime numbers

  2. One composite number and one prime number can also form a pair of co-prime numbers

  3. Two composite number can also form pair of co-prime numbers but not always.

  4. Two consecutive natural numbers are always co-prime.

Factors and Multiples

  1. Factors of any given number will always be less than or equal to the original number.

  2. Multiples of any given number will always be greater than or equal to the original number

  3. If the remainder is zero and you want another one then increase the original number by the remainder you want.

  4. If the remainder is not zero and to make it zero subtract the remainder from the original number.

Fundamental Theorem of Arithmetic

Every composite number can be expressed as product of its prime factorisation , in a unique way other than the order of the factors.

  1. HCF = product of the common factors with the lowest exponential powers

  2. LCM = product of all the prime factors with the highest exponential powers

For any given 2 numbers , the product of 2 numbers is always equal to the product of their HCF and LCM. This formula is only valid for 3 numbers

a x b = HCF x LCM (a,b)

Conditions

A - Composite Number

If a number can be expressed as a product of prime factors it follows the fundamental theorem of arithmetic and is composite or if a number has more than 2 factors it is composite.

B - Units Place is 0

For any numbers to have a unit place zero the number must contain the prime factors 2 and 5 both in the prime factorisation irrespective of their powers. It can still contain prime factors other than 2 and 5.

Points to Remember

  1. For verification questions do not solve them simultaneously

  2. Always write prime factorisation as the product of powers

  3. For checking composite for the form of questions a x b x c x d + k , try to take common first.

  4. For units place zero always write the we know that a it need to contain both 2 and 5 and then write the therefore statement , the expansion also carries one mark.

  5. If p is a prime number that divides a square then p also divides a