Lesson 8ª

HIGHEST COMMON DENOMINATOR

Highest common denominator

Pay attention to the definition of each one of the 3 words: highest, common, denominator.

Highest: The highest number.
Common: That works for two or more numbers.
Denominator: That when dividing by this number, the remaider of each one of the divisions is zero.

Example:

We have the numbers 24 and 20 Which is the highest number we can write as denominator that when dividing 24 and 20 by that number gives us a remainder of zero?
What number would you write instead of X?

Since 24 and 20 are divisible by 2, we could write 2 instead of X.

Let's see if we could also replace X by 4.
In both cases, the remainder is zero.
Which one of both numbers is the HIGHEST COMMON DENOMINATOR?
As we know, the highest would be 4.
The highest common denominator for 24 and 20 is 4 so we write it in a simpler form:

h.c.d. (24,20) = 4

3.51 Which is the highest common denominator for 15 and 3?

3.52 Which is the highest common denominator for 21 and 14?

3.53 Which is the highest common denominator for 30 and 20?

3.51 The h.c.d. (15 and 10) = 5

3.52 The h.c.d. (21 and 14) = 7

3.53 The h.c.d. (30 and 20) = 10

Can we calculate the h.c.d. for more than two numbers?
We can calculate it for three or more numbers.

3.54 Let's calculate:

The h.c.d. for (90, 36 and 12)

Calculating this number in your head or testing requires a lot of work.
There are several ways of doing this. We will study two ways of doing this without using a computer. You may choose the one which suits you better.

We will break down the numbers 90, 36 and 12 into their prime factors:

When we have finished breaking the numbers down into their prime factors, we write:

Now, we have to notice which factor or denominator with a lower exponent is present in all three numbers.
Both 2 and 3 are among the factors in 90, 36 and 12. Then, 2x3=6, which is the highest common denominator. So, we write:

h.c.d. (90, 36 and 12) = 6

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