E.g. 3, 6, 9, 12, 15, 18, ...... are multiples of 3.

·       1 is a factor of every number.

·       Every number is a factor of itself.

·       Every factor of a number is an exact divisor of that number.

·       Every factor of a number is less than or equal to the number.

·       Factors of a given number are finite.

·       Every multiple of a number is greater than or equal to that number.

·       Every number is multiple of itself.

·       The number of multiples of a given number is infinite.

E.g., the factors of 6 are 1, 2, 3 and 6.

(i) Divide the number by 2 if the number is divisible by 2. Find the quotient and list 2 and the other factor (quotient) just below 2.

(ii) Divide the number by 3 if the number is divisible by 3. Find the quotient and list 3 and the quotient just below 3.

(iii) Divide the number by 4 if the number is divisible by 4. Find the quotient and list 4 and the quotient just below 4.

Continue this process with 5, 6, 7, etc. till we get a factor which has already been listed.

(i) 2 is the smallest prime number.

(ii) 2 is the only even prime number. All other even numbers are composite numbers.

(iii) If a number is not divisible by any one of the primes less than half of it, then it is prime. Otherwise it is a composite number.

3,5; 5,7; 11,13; 17,19; 29,31; 41,43; 59,61 and 71,73.

The only prime triplet is (3, 5, 7).

2,3; 3,4; 5,6; 8,13; 12,23, etc. are pairs of co-primes.

(i) There are infinitely many primes

(ii) Every prime number except 2 is an odd number.

(iii) A natural number greater than 1 is either a prime or it can be expressed as a product of primes.

(iv) There is no largest prime number.

(v) If p and q are prime factors of a number a, then their product p × q is also a factor of a.

(i) Choose the smallest prime p which divides n. Divide n by p. Let the quotient be m₁.

(ii) If m₁ is prime, then p × m₁ is the prime factorization of n. Otherwise, we choose the smallest prime q which divides m₁. Let the quotient be m₂.

(iii) If m₂ is prime, then p × q × m₂ is the prime factorization of n. Otherwise we continue the same process till we get a prime quotient.

A number is divisible by 10, if its unit’s digit is zero.

A number is divisible by 5, if its unit’s digit is either 0 or 5.

A number is divisible by 2, if its unit’s digit is 0, 2, 4, 6 or 8.

A number is divisible by 3 if the sum of its digit is divisible by 3.

A number is divisible by 9, if the sum of its digits is divisible by 9.

A number is divisible by 4, if the number formed by its digits in ten's and unit's places is divisible by 4.

A number is divisible by 6, if it is divisible by both 2 and 3.

A number is divisible by 8, if the number formed by its digits in hundred's, ten's and unit's places is divisible by 8.

A number is divisible by 11, if the difference of the sum of its digits in odd places and the sum of its digits in even places (starting from unit's place) is either 0 or a multiple of 11.

(i) If a number is divisible by another number, then it is divisible by each of the factors of that number.

OR

If a, b, c are three natural numbers such that a is divisible by b and b is divisible by c, then a is divisible by c also.

E.g.,

·       Since 4 is divisible by 2. Therefore, every number divisible by 4 is also divisible by 2.

·       Since 6 is divisible by 2 and 3 both. Therefore, every number divisible by 6 is also divisible by both 2 and 3.

(ii) If a number is divisible by each of the two or more co-prime numbers, then it is divisible by their products.

OR

If a and b are two co-prime numbers such that a number c is divisible by both a and b, then c is also divisible by a × b.

E.g.,

·       Since two prime numbers are always co-prime, therefore, if a number is divisible by each one of any two prime numbers, then the number is divisible by their product.

·       Since 2 and 3 are co-primes. Therefore, if a number is divisible by both 2 and 3, it must be divisible by 2 × 3 = 6.

(iii) If a number is a factor of each of the two given numbers, then it is a factor of their sum.

OR

If two numbers b and c are divisible by a number a, then b + c is also divisible by a.

(iv) If a number is a factor of each of the two given numbers, then it is a factor of their difference.

OR

If two numbers b and c are divisible by number a, then b – c is also divisible by a.

(i) Prime factorization method

(ii) Continued division method

(i) Write prime factorization of the given numbers.

(ii) Identify common prime factors.

(iii) For each common prime factor find the minimum number of times it occurs in the prime factorizations of the given numbers.

(iv) Multiply each common prime factor the number of times determined in Step (iii) and find their product to get the required HCF.

(i) Take the greater number as dividend and the smaller one as divisor.

(ii) Find the quotient and remainder.

(iii) If the remainder is zero, then divisor is the required HCF. Otherwise go to next step.

(iv) Take the remainder as the new divisor and the divisor as the new dividend.

(v) Repeat steps (ii), (iii) and (iv) till the remainder zero is obtained. The last divisor, for which the remainder is zero, is the required HCF.

(i) Find the HCF of any two of them.

(ii) Find the HCF of the third number and HCF obtained in Step (i).

(iii) Take the HCF obtained in Step (ii) as the required HCF of three given numbers.

(i) Obtain a common factor of the given numbers.

(ii) Divide the numbers by the common factor obtained in Step (i) and write the resultant quotients.

(iii) Obtain a common factor of the resultant quotients obtained in Step (ii).

(iv) Repeat Step (ii) and (iii) until there are no common factors left.

(v) Find the product of all the common divisors obtained in the above steps. This product gives the HCF of the given numbers.

(i) Prime factorization method.

(ii) Common division method

(i) Write prime factors of each of the numbers and express them in exponential form.

(ii) Find the product of all different prime factors with highest power in the prime factorization of each number.

(iii) The number obtained in Step (ii) is the required LCM.

(i) Arrange the given numbers in a row separated by commas.

(ii) Obtain a number which divides exactly at least two of the given numbers.

(iii) Divide two chosen numbers by the number obtained and write the quotients just below them. Carry forward the numbers which are not divisible.

(iv) Repeat Step (ii) and (iii) till no two of the given numbers are divisible by the same number.

(v) Find the product of the divisors and the undivided numbers to get the required LCM of the given numbers.

(i) the least number that should be subtracted from ' a' so that 'b' divides the difference exactly = r.

(ii) the least number that should be added to 'a' so that 'b' divides the sum exactly = b – r.

(i) The HCF of given numbers is not greater than any of the numbers.

(ii) The LCM of given numbers is not less than any of the given numbers.

(iii) The HCF of two co-prime numbers is 1.

(iv) The LCM of two or more co-prime numbers is equal to their product.

(v) If a number, say x, is a factor of another number, say y, then the HCF of x and y is x and their LCM is y.

(vi) The HCF of given numbers is always a factor of their LCM.

(vii) The product of the HCF and the LCM of two numbers is equal to the product of the given numbers. That is, if a and b are two numbers, then