Factors and Multiples

Basics Concepts

• Factors and Multiples

A number P is exactly divisible by Q i.e. when P is divided by Q, the remainder is 0.

Then:  Q is called a factor of P

P is called a multiple of Q

e.g. 8 is the factor of 24 and 24 is the multiple of 8

(24 = 1 x 2 x 2 x 2 x 3)

Hence, the factor of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

• Highest Common Factor (HCF)

Highest Common Factor is largest integer that exactly divides two or more given numbers.

It is also known as Greatest Common Divisor (GDC) or Greatest Common Measure (GCM)

Factorization method to find HCF

Express each given number as the product of primes. The product of the prime factors common to all the numbers will be the required to HCF.

Find the HCF of 342, 658, 986.

e.g. if we factorise the numbers, we get

324 = 2 x 3²x 19

658 = 2 x 7 x 47

986 = 2 x 17 x 29

Hence, HCF of 324, 658, 986 is 2

• Least Common Multiple (LCM)

The least number which is exactly divisible by each of the given numbers is called their Least Common Multiple (LCM).

Factorization method to find LCM

Resolve each one of the given numbers into prime factors. Their LCM is the product of the highest powers of the factors that occur in these numbers.

e.g. Find the LCM of 24,54 and 70

24 = 2 x 2 x 2 x 3 = 2³ x 3

54 = 2 x 3 x 3 x 3 = 2 x 3³

70 = 2 x 5 x 7

LCM =  2³x 3³ x 5 x 7 = 7560

Application of Basic Concepts (Factors and Multiples)

1. Product of numbers = HCF x LCM
2. To find the greatest number that will exactly divide x, y and z = HCF of x, y and z.
3. To find the least number which is exactly divisible by x, y and z = LCM of x, y and z.

• If a, b, c…. are positive integer that divide a number ‘n’ to leave the same remainder ‘r’, then the smallest value of n=r
• If a, b, c…. are positive integer that divide a number ‘n’ to leave different remainders p, q, r respectively such that (a – b) = (b – q) = (c – r) = d, then the smallest value of n = (LCM of a, b, c….) -d

4. The greatest number that will exactly divide x, y and z leaving remainders a, b and c respectively = HCF of (x – a), (y – b) and (z – c).
5. The greatest numbers that will exactly divide x, y and z leaving the same remainder in each case = HCF of (x – y) (y – z) (z – x).

Important concept to note.

• LCM is always greater than or equal to the greatest of the given numbers.
• HCF is always equal to or less than the least of the given numbers.
• HCF of the given number is always the factor of their LCM.

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