HCF (Highest common factors)
When 2 or more numbers are increased together, the individual numbers are known as factors. Thus, a factor is a number that divides into another number exactly. The highest common factor (HCF) is the largest variety that divides into 2 or more numbers specifically.
For the sample, study the numbers 12 and 15.
- The factors of 12 are 1, 2, 3, 4, 6, and 12 (i.e. all the numbers that are divided into 12).
- The factors of fifteen are 1, 3, 5, and 15 (i.e. all the numbers that are divided into 15)
- 1 and 3 are the sole common factors; i.e., numbers that are factors of each 12 and 15.
Therefore, the HCF of twelve and fifteen is 3 since 3 is the highest number that divides into each 12 and 15.
BASIC ARITHMETIC – MATHEMATICS
Lowest Common Multiple (LCM)
A multiple may be a number that contains another number a definite 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 the sample, study 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 each 12 and 15) and there are not any lower common multiples.
Hence, the least common multiple (LCM) of 12 and 15 is 60 since 60 is that the lowest number that each 12 and 15 divide into.
Here are some further problems involving the determination of HCF and LCM.
Problem 1. Define 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 prime numbers 2, 3, 5, 7, 11, 13, (where possible) successively. Thus,
⇓ ⇓ ⇓ ⇓
12 = 2 × 2 × 3
30 = 2 × 3 × 5
42 = 2 × 3 × 7
The factors that are common to every one of the numbers are 2 in columns 1 {and 3|and three} 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 is able to divide into 12, 30, and 42.
Problem 2. Define 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. Define 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 least common multiple (LCM) is 2 × 2 × 3 × 3 × 5 × 7 = 1260 and is the smallest number that 12, 42, and 90 can all divide into specifically.
Problem 4. Define 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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