Concepts of Factor and Multiple (HCF & LCM)
When two or more numbers are multiplied together to give a product, each is called a factor of the product and the product is said to be a multiple of each of those numbers.
e.g.
HCF (Highest Common Factor)
It is also known as GCD (Greatest Common Divisor) and GCF (Greatest Common Factor). The HCF of two or more numbers is the largest of their common factors.
e.g.
Factors of 15 are 1, 3, 5 and 15.
Here, greatest and common factor of 10 and 15 is 5.
Therefore, their HCF is 5.
Methods of Finding HCF
1. Factorization Method
Express each of the given numbers as the product of their prime factors. The product of least power of common prime factors gives HCF.
Example. Determine the HCF of 36 and 64.
Solution
36 = 2 x 2x 3x 3 = 22 x 32
64 = 2 x 2 x 2 x 2 x 2 x 2 = 26
Here, least powers of prime factors are 22 and 30.
Therefore, HCF = 22 = 4
2. Division Method
Divide the large number by the smaller number. Now, divide the divisor by the remainder. Repeat this process until, we get zero as remainder. The last divisor is the required HCF.
Example. Determine the HCF of 72 and 48.
Solution
Therefore, HCF = 24
LCM (Least Common Multiple)
LCM of two numbers is the smallest number which is a multiple of both the numbers.
e.g.
Multiples of 4 are 4, 8, 12, 16, 20, 24,…
Here, smallest multiple of 6 and 4 is 12.
Therefore, their LCM is 24.
Methods of Finding LCM
1. Factorization Method
Express each of the given numbers as the product of their prime factors. The product of all the prime factors of each of the given number with greatest index of common prime factors gives the LCM.
Example. Find the LCM of 72, 108 and 210.
Solution
108 = 22 x 33
210 = 2 x 3 x 5 x 7
Here, highest power of all factors 23, 33, 5 and 7.
Therefore, LCM = 23 x 33 x 5 x 7 = 7560
2. Division Method
Arrange the given numbers in a row separating them by commas and divide by a number which divides exactly atleast two of the given numbers and write their quotient in the next row. Also, carry down the numbers which are not divisible. Repeat this process till no two of the numbers are divisible by same number except one. The product of the divisors and the undivided number is the required LCM.
Example. Find the LCM of 16, 24, 35 and 42.
Solution
Therefore, LCM = 2 x 2 x 2 x 3 x 7 x 2 x 5 = 1680
Important Formulae
Problem Solving Tips
Q. Find the greatest number that will divide x, y and z exactly.
Tip. The required number is the HCF of x, y and z.
Q. Find the greatest number that will divide x, y and z leaving remainders a, b and c, respectively.
Tip. The required number is the HCF of (x-a), (y-b) and (z-c).
Q. Find the smallest number which is exactly divisble by x, y and z.
Tip. The required number is the LCM of x, y and z.
Q. Find the smallest number which when divided by x, y and z leaves the remainders a, b and c, respectively.
Tip. The required number is the LCM of (x-a), (y-b) and (z-c).
