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Lesson 4: The Highest Common Factor (H.C.F) and the Lowest Common Multiple (L.C.M)
FACTORA
factor
of a given number is every number that
divides exactly
into that number. ExampleWrite down all factors of 10. 10 = 2 x 5, so numbers 2 and 5 are factors of 10. The factors of 10 are 1, 2, 5, 10. NOTE: Number 1 and the number itself are always factors of any number. PRIME AND COMPOSITE NUMBERS.A prime number
has exactly 2 factors,
the number itself
and
1. Example: 5 is a prime number, because the only factors it has are 1 and 5. The prime numbers less than
20 are 2,3,5,7,11,13,17,19 Example:Find all prime factors of 30. Solution: All the factors of 30 are 30, 15, 10, 6, 5, 3, 2, 1 A composite
number
has
at least one more factor that the number
itself or 1. The composite numbers less than 20 are 4, 6, 8, 9, 10,
12, 14, 15, 16, 18 DIVISIBILITY RULESThe simple divisibility rules will help you to find factors of a number. The number is divisible by:
NOTE: If a number is divisible by 2 factors, it is also divisible by the product of these factors. Example 1: Number 18 is divisible by 2 and 3, so it must be divisible by 2 x 3 = 6. Example 2: Number 945 is divisible by 9 (why?)
and by 5 (why?), so it must be divisible by 9 x 5 = 45. (Can you check it?)
COMMON FACTORSWhen two (or more) numbers have the same factor, that factor is called a common factor. ExampleFind all the common factors of 12 and 18. Factors of 12 are 1,
2,
3, 4,
6, 12. HIGHEST COMMON FACTOR (H.C.F).The Highest Common
Factor (H.C.F)
of two (or more) numbers is the
largest number that divides evenly
into both numbers.
The common factors or of 12 and 18 are 1, 2, 3 and 6.
It is very easy to find
a H.C.F. of small numbers, like 6 and 9 (it is 3) or 8 and 4 (it is
4). FINDING THE H.C.F. OF BIG NUMBERSFor larger numbers you can use the following method:
Example:Find the Highest Common Factor (H.C.F.) of 240 and 924. Solution: Finding all prime factors of 240: We will start with the smallest prime number
and we will divide 240 into it if we can (the
divisibility rules come handy).
NOTE: There are other methods for finding all prime factors of a number,
for example a factor tree.
924 = 2 x 2 x 3 x 7 x 11 Multiply the factors which repeat in both numbers to get the H.C.F. The Highest Common Factor is 2 x 2 x 3 = 12 MULTIPLES. COMMON MULTIPLES.When you multiply a given whole number by any
other whole number, the result is a
multiple
of that number. Example 2: The common multiples of two numbers are multiples of both numbers. Example 1: LOWEST COMMON MULTIPLE (L.C.M.).The Lowest Common Multiple (L.C.M) is the smallest number that is a common multiple of two or more numbers. For example, the L.C.M of 3 and 5 is 15 .(see the
example above). Example 1:Find the Lowest Common Multiple of 8 and 12. Solution: Multiples of 12 are 12, 24...
FINDING L.C.M. OF BIG NUMBERS
Example:Find the Lowest Common Multiple (L.C.M.) of 240 and 924. From the example of finding the H.C.F. we know the prime factors of both numbers. 924 = 2 x 2
x 3 x 7 x 11 The L.C.M. is 2 x 2 x 3 x 7 x 11 x 2 x 2 x 5 = 924 x 20 = 18,480


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