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Highest Common Factor (H.C.F.)

by Urwa
(Saudi Arabia/Makkah)











































HCF and LCM

   • Please explain (in detail with pictures):

           HCF (highest common factor)

           LCM (lowest common multiple)

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Nov 19, 2012
HCF and LCM
by: Staff


Answer


Part I

The Least Common Multiple

To understand what the LCM is, look at the following example.

Find the Lease Common Multiple of 5 and 7.
1. Find all the “multiples” of 5 and 7

Find all the multiples of 5

There are many multiples of the number 5.

Multiples of 5 are:

           5 * 1 = 5

           5 * 2 = 10

           5 * 3 = 15

           5 * 4 = 20

           5 * 5 = 25

           5 * 6 = 30

           5 * 7 = 35

           5 * 8 = 40

           5 * 9 = 45
              .
              .
              .
           and so on


Find all the multiples of 7

There are many multiples of the number 7.

Multiples of 7 are:

           7 * 1 = 7

           7 * 2 = 14

           7 * 3 = 21

           7 * 4 = 28

           7 * 5 = 35

           7 * 6 = 42

           7 * 7 = 49

           7 * 8 = 56

           7 * 9 = 63
              .
              .
              .
           and so on


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Nov 19, 2012
HCF and LCM
by: Staff

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Part II


2. Find all the “common multiples” of 5 and 7

Are any of the multiples of five the same as any of the multiples of 7

Yes.

           both
           5 * 7 = 35 and 7 * 5 = 35

           both
           5 * 14 = 70 and 7 * 10 = 70

           both
           5 * 21 = 105 and 7 * 15 = 105

           both
           5 * 28 = 140 and 7 * 20 = 140

           both
           5 * 35 = 175 and 7 * 25 = 175
              .
              .
              .
           and so on

All the common multiples of 5 and 7 are:

           35, 70, 105, 140, 175, . . . 35n, where n is a natural number



3. Find the “Least Common Multiple” of 5 and 7

The LCM is the smallest common multiple.

LCM of 5 and 7 = 35

In summary:

LCM: The least common multiple of two or more numbers is three things:

     1) first, a MULTIPLE;
     2) second, a COMMON MULTIPLE; and
     3) last, the LEAST OF ALL the possible COMMON MULTIPLES.


How is the LCM used?

The most common application is adding and subtracting fractions.

Suppose you wanted to add the following fractions:

1/5 + 1/7

As you know, you cannot add or subtract fractions unless those fractions have the same denominator.

You can convert both fractions any common multiple. No matter which multiple you decide to use, the final answer will be the same.

1/5 + 1/7

Using the common multiple of 35
1/5 + 1/7 = 7/35 + 5/35 = 12/35

Using the common multiple of 70
1/5 + 1/7 = 14/70 + 10/70 = 24/70 = 12/35

Using the common multiple of 105
1/5 + 1/7 = 21/105 + 15/105 = 36/105 = 12/35
.
.
.
and so on



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Nov 19, 2012
HCF and LCM
by: Staff


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Part III


However, if you use the Least Common Multiple (35) as a common denominator, there is simply a lot less work involved in adding the fractions.


The LCM is also used for a variety of other problems.

For example, suppose you plan to provide guests with ice cream and strawberries at a private social gathering. You plan to give each guest one scoop of ice cream with one strawberry on top.

1 pint of ice cream contains 5 scoops of ice cream.

1 small box of premium strawberries contains exactly 7 strawberries.

To ensure nothing is wasted, the number of scoops must equal the number of strawberries.

The LCM is 35.

You must buy 7 pints of ice cream (7 * 5 = 35 scoops) for every 5 small boxes of strawberries (5 * 7 = 35 strawberries).

There are many multiples for every number.

There will be many common multiples for any two (or more) numbers.

But, . . . there will be only one LCM.


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The Highest Common Factor (also called the Greatest Common Factor)

The HCF (Highest Common Factor) is the largest common factor of the two numbers.

To find the HCF, begin by listing all the factors of each number.

When a factor appears as a factor in both lists, it is a common factor.

The GCF is computed by multiplying all the common factors.

For example, the factors of 30 and 18 are listed below:


30 = 1 x 2 x 3 x 5

18 = 1 x 3 x 3 x 2


The common factors for the numbers 18 and 30 are 1, 2 and 3.

Therefore, the greatest common factor is “1 x 2 x 3 = 6”.

30 = 5 x 6

18 = 3 x 6


In summary:

GCF: The greatest common factor of two or more numbers is three things:

     1) first, a FACTOR

     2) second, a COMMON FACTOR

     3) last, the GREATEST OF ALL the COMMON FACTORS.



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Nov 19, 2012
HCF and LCM
by: Staff


------------------------------------------------
Part IV


So why does your teacher want you to know about the GCF?

There is a reason. You are learning a basic skill which you will need.

Although your math coursework has not covered this topic yet, in the future your ability to use the GCF will help you to factor complicated polynomial expressions such as 4x⁴+20x³+8x².


The GCF is also used for practical, everyday problems.

For example, suppose you are buying square tiles for the floor of a room which measures 10 ft by 11 ft.

You don’t want to cut any of the tiles if you can avoid it.

What is the size of the largest whole, square tile that can be used to completely cover the floor?

To solve this problem, compute the Greatest Common Factor for the length and width of the room.

Length = 11 ft x 12 in per foot = 132 inches

Width = 10 ft x 12 in per foot = 120 inches

Factors of 132 = 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132

Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

The GCF is 12.

The length of the largest square tile that can fit into this room is 12 in by 12 in.

You can use exactly 110 uncut tiles to cover the floor. No tile needs to be cut.


There may be many factors for a number.

There may be many common factors for any two (or more) numbers.

But, . . . there will be only one GCF.






Thanks for writing.

Staff
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