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HCF AND LCM – MATHEMATICS

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HCF

HCF (Highest common factors)

When two or more numbers are multiplied together, the individual numbers are called factors. Thus, a factor is a number that divides into another number exactly. The highest common factor (HCF) is the largest number which divides into two or more numbers exactly.

For example, consider the numbers 12 and 15.

  • The factors of 12 are 1, 2, 3, 4, 6 and 12 (i.e. all the numbers that divide into 12).
  • The factors of 15 are 1, 3, 5 and 15 (i.e. all the numbers that divide into 15)
  • 1 and 3 are the only common factors; i.e., numbers which are factors of both 12 and 15.

Hence, the HCF of 12 and 15 is 3 since 3 is the highest number which divides into both 12 and 15.

BASIC ARITHMETIC – MATHEMATICS

Lowest Common Multiple (LCM)

A multiple is a number that contains another number an exact number of times. The smallest number which is exactly divisible by each of two or more numbers is called the lowest common multiple (LCM).

For example, consider the numbers 12 and 15.

  • The multiples of 12 are 12, 24, 36, 48, 60,  72,. . .
  • The  multiples of 15  are  15,  30,  45, 60, 75,. . .
  • 60 is a common multiple (i.e. a multiple of both 12 and 15) and there are no lower common multiples.

Hence, the LCM of 12 and 15 is 60 since 60 is the lowest number that both 12 and 15 divide into.

Here are some further problems involving the determination of HCF and LCM.

Problem 1. Determine the HCF of the numbers 12,30 and 42

Probably the simplest way of determining an HCF is to express each number in terms of its lowest factors. This is achieved by repeatedly dividing by the prime numbers 2,3,5,7,11,13, … (where possible) in turn. Thus,

                                                                                 ⇓    ⇓      ⇓    ⇓

12 = 2 × 2 × 3

30 = 2       × 3 × 5

42 = 2       × 3 × 7

The factors which are common to each of the numbers are 2 in column 1 and 3 in column 3, shown by the broken lines. Hence, the HCF is 2 × 3; i.e., 6. That is, 6 is the largest number which will divide into 12, 30 and 42.

Problem 2.  Determine the HCF of the numbers 30,105,210 and 1155

Using the method shown in Problem 1:

                                                      ⇓          ⇓    ⇓      ⇓    ⇓

 30 = 2      × 3 × 5

105 =           3 × 5 × 7

210 = 2    × 3 × 5 × 7

1155 =        3 × 5 × 7 × 11

The factors which are common to each of the numbers are 3 in column 2 and 5 in column 3. Hence, the HCF is 3 × 5 = 15.

Problem 3.   Determine the LCM of the numbers 12,42 and 90

The LCM is obtained by finding the lowest factors of each of the numbers, as shown in Problems 1 and 2 above, and then selecting the largest group of any of the factors present. Thus,

12 = 2 × 2  × 3

42 = 2         ×  3                  × 7

90 = 2         ×  3 × 3  × 5

The largest group of any of the factors present is shown by the red color and is 2 × 2 in 12, 3 × 3 in 90, 5 in 90 and 7 in 42.

Hence, the LCM is 2  × 2 × 3 × 3 × 5 × 7 = 1260 and is the smallest number which 12, 42 and  90 will all divide into exactly.

Problem 4.   Determine the LCM of the numbers 150,210,735 and 1365

Using the method shown in Problem 3 above:

150 =  2  ×  3 × 5 × 5

210 = 2  × 3  × 5         ×  7

735 =           3  × 5        ×  7 × 7

1365 =           3  × 5        ×  7         ×  13

Hence, the LCM is  2 × 3 × 5 × 5 × 7 × 7 × 13 = 95550.

REVISION OF MULTIPLICATION AND DIVISION

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