Divisibility

DIVISIBILITY

Any Interger I, when divided by a natural  number N, there exist a quotient Q and reminder R,
respectively.

I = QN + R

Theorems of divisibility:
1. If a is divisible by b then a.c is also divisible by b.
2. If a is divisible by b, b is divisible by c then a is divisible by c.
3. If a & b are natural numbers such that a is divisible by b and b is divisible by a then a = b.
4. If n is divisible by d and m is divisible by d then (m+n) and (m-n) also divisible by d.
      ex: 28, 742 both are divisible by 7, hence (28 + 742) and (28 - 742) are divisible by 7.

HCF & LCM

RULES FOR FINDING HCF OF TWO OR MORE NUMBERS

School Process:
a). Find the standard form of numbers n1 & n2. Standard form:- Finding the prime factors of the number.  example:  80 = 2 x 2 x 2 x 2 x 5 = 24 x 51.
b). write all prime factors that are common to both standard forms of numbers n1 & n2 which having lesser powers.c). The product of these prime factors will be the GCD of n1 and n2.
example: GCD of 25, 35
25 = 52, 35= 5 1 x 71
same prime factors = 5, Hence HCF will be 5.

SHORT CUT FOR FINDING HCF/GCD

1. First find the smallest difference between any pair of numbers.
2. Then find the factors for the difference. One of these factor numbers has to be the HCF of the Numbers.
3. For that we have identify the factor which divides all numbers in Question.
4. example:
   39,78 and 195
   i.  Differnce b/w 78-39 = 39
   ii. factors for 39 are: 1, 3, 13, 39
   iii. 39  is HCF, because all three numbers divisible by 39.

RULES FOR FINDING LCM (Least Common Multiple) OF TWO OR MORE NUMBERS.

 School Process:
example: Find the LCM of 150, 210, 375

Step 1: Find the standard form of the numbers 150, 210 and 375.

            150 = 5 x 5 x 3 x 2

            210 = 5 x 2 x 7 x 3

            375 = 5 x 5 x 5 x 3

Step 2: Write down all prime factors; that appears at least once in any of the numbers: 2, 3, 5, 7.

Step 3: Raise each of the prime factors to their highest available power from the standard forms written above.

     The LCM = 2¹ x 3¹ x 5³ x 7¹ = 5250.

Important Rule:
GCD(n1, n2) * LCM(n1,n2) = n1*n2
i.e., The product of the HCF and the LCM equals to the product of the numbers.

CO-PRIMES

Some rules for Co-Primes
2 Numbers being Co-Prime

• Two consecutive natural numbers are always co-primes (ex: 5,6; 82, 83;)
• Two consecutive odd numbers are always co-primes (ex: 7,9; 51,53;)
• Two prime numbers are always co-primes (ex: 13,17; 53,73)
• One prime number & another composite number (composite is not multiple of the prime number) are always co-primes.
  But note that 17 and 51 are not co-primes

3 Numbers being Co-Prime

• Three consecutive odd numbers are always co-primes (ex: 15,17,19; 82, 83;)
• Three consecutive natural numbers with the first one being odd. (ex: 15,16,17; 21,22,23)
• Two consecutive natural numbers along-with the next odd number. (ex: 22, 23, 25; 52,53, 55;)
• Three prime numbers are always co-primes.

RULES FOR FINDING HCF & LCM OF FRACTIONS

1. HCF of the two or more fractions is given by : HCF of Numerators / LCM of Denominators.
2. LCM of the two or more fractions is given by : LCM of Numerators / HCFof Denominators.

Prime Numbers

Prime Number: A natural number Greater than UNITY(1) is a prime if it doesn’t have other divisor / factor except for itself and UNITY.

Note: Unity (i.e. 1 ) is not prime number.

Properties of prime numbers: 

1. Lowest prime number is 2.
2. 2 is also the one & only even prime number.
3. The lowest odd prime number is 3.
4. The reminder of the division of the square of a prime number ‘p’
   (p > 3) divided by 12 or 24 is “ONE(1)”.
   • eg 1: p = 13
     • p 2 = 169 (169 > 3)
     • 169 / 12 , 169 / 24 both will give remainder ‘1’.

Short cut to check a number is prime or not:

1. Take the square root of the given number(N).
2. Round of the square root to the immediate lower integer. (assume it is z)
3. Check for divisibility of the number N by all prime numbers below z.
   If there is no prime number below z which divides N then the number N will be prime.
4. example: 113
   Square root of 239:  √113  lies between 10 and 11. Hence take z as 10.
   primes below 15 are 2,3,5 & 7. 113 is not divisible by any of these.
   So we can conclude that 113 is a prime number.