Factors and Multiples -Factors, Common Factors, Highest Common Factors (H.C.F.)

Factors and Multiples -Factors, Common Factors, Highest Common Factors (H.C.F.)

and

Multiples, Common Multiples, Lowest Common Multiples (L.C.M.)

We know that the exact divisor of a number is called its factor. For example 10 is exactly divisible by 5 and so, we can say that 5 is the factor of 10.

 Let us consider some numbers, say 8, 12 and 20

Now, we have

Common Factors of 8, 12 and 20 are 1, 2 and 4

Highest Common Factor of 8, 12 and 20 is 4

 4 is the largest number which divides 8, 12 and 20 that is, 4  is largest exact divisor of 8,12 and 20 and so, we can say that HCF of 8,12 and 20 is 4

 Hence,

The largest number which divides exactly two or more numbers is called the Highest Common Factors (H.C.F.) of the numbers.

Or

The largest number which is the exact divisor of two or more numbers is called the Highest Common Factors (H.C.F.) of the numbers.

 Let us consider some numbers, say 4, 6 and 8

Then, we have

From above, we see that

Now, common multiples of 4, 6 and 8 are 24, 48 and 72

Lowest Common Multiple of 4, 6 and 8 is 24 and so, we can say that L.C.M. of 4, 6 and 8 is 24

In other words, we can say that 24 is the smallest number which is exactly divisible by 4, 6 and 8

Hence,

The smallest number which is exactly divisible by two or more numbers is called the Lowest Common Multiple (L.C.M.) of the numbers.

RELATION BETWEEN TWO NUMBERS

The Product of two numbers is equal to the Product of the HCF and LCM of the two numbers. This is the mathematical relation between two numbers.

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Product of two numbers = Product of the HCF and LCM of two numbers

 

Let us understand this concept by taking two numbers, say 8 and 12

 

Here, we have

HCF of 8 and 12 = 4

LCM of 8 and 12 = 24

From (i) and (ii), we get

8 × 12 = HCF of 8 and 12 × LCM of 8 and 12

 

Hence, it is well established that

Product of two numbers = Product of the HCF and LCM of two numbers

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