Prime numbers

These are numbers whose factors are ONLY 1 and themselves. If they are divided by any other number, the answer will not be a whole number. An example is 11.

11 ÷ 1 = 11                  11 ÷ 5 = 2.2                 11 ÷ 9 = 1.22
11 ÷ 2 = 5.5               11 ÷ 6 = 1.83               11 ÷ 10 = 1.1
11 ÷ 3 = 3.67            11 ÷ 7 = 1.57                11 ÷ 11 = 1
11 ÷ 4 = 2.75            11 ÷ 8 = 1.38

Since only 1 and 11 are the factors of 11, it is said to be a prime number. Others are 3, 5, 7, 13, 17, 19, 23, …
Now, the HCF of two numbers one of which is a prime number is 1.

For instance, what is the HCF of 5 and 6?
Factors of 5 = 1 and 5
Factors of 6 = 1, 2, 3 and 6
Common factor of 5 and 6 = 1
HCF of 5 and 6 = 1

The rule also applies if both numbers are prime numbers.
Example, what is the HCF of 13 and 17?
Factors of 13 = 1 and 13
Factors of 17 = 1 and 17
Common factor of 13 and 17 = 1
HCF of 13 and 17 = 1

Observe that whenever the common factor of a set of numbers is 1, the HCF of the numbers is also 1. This is normal Mathematics. But the rules of Inheritance Arithmetic which I call “inherithmetic” sometimes violate well known Mathematical principles. The most important of these violations is that inherithmetic DOES NOT recognise 1 as a common factor. So, revisiting our earlier solutions,

Common factor of 10 and 6 = 2
Common factor of 12 and 15 = 3
Common factor of 5 and 6 = No common factor!
Common factor of 13 and 17 = No common factor!

In order to differentiate between common factor of Mathematics which incorporates 1 and the common factor of inherithmetic that does not recognise 1, the latter will be renamed “Common Divisor” and henceforth, that is what will be used. As the name implies, common divisor is a number OTHER THAN 1, that can divide the numbers under consideration and the answers will be whole numbers. In case there exist 2 or more common divisors, the “Highest Common Divisor (HCD)” is used.

Recall that two or more numbers are parallel when they are not the same and one is not a multiple of the other. Also, the LCM of parallel numbers is gotten by simply multiplying them. At this point, this method of finding the LCM of parallel numbers will be modified. The modification does not affect what has being discussed earlier. The new rule is: if two parallel numbers ‘A’ and ‘B’ HAVE A COMMON DIVISOR, their LCM is determined by DIVIDING ‘A’ with the common divisor, then use the solution to MULTIPLY ‘B’. Alternatively, divide ‘B’ by the common divisor and multiply the solution with ‘A’. Both approaches will give the same answer. But when the parallel numbers HAVE NO COMMON DIVISOR, the previous rule of multiplying them gives the LCM.

Question: What is the LCM of 3 and 7?
Common divisor of 3 and 7 = None
LCM of 3 and 7 = 3 × 7 = 21

Question: What is the LCM of 6 and 8?
Common divisor of 6 and 8 = 2
LCM of 6 and 8 = 6 ÷ 2 = 3 × 8 = 24   or   8 ÷ 2 = 4 × 6 = 24

Notice that if the previous rule were applied, the LCM should have been 6 × 8 = 48; which is not quite correct. Let’s prove it.

Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …
Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …
Common multiples of 6 and 8 = 24, 48, 72, 96, 120, …
LCM of 6 and 8 = 24

Therefore, it can be established that not all parallel numbers are actually parallel. Some tend to “converge” at a point. This phenomenon is called TAWAFUQ, which I translate as Converge. In summary, the 4 relationships between numbers are:

i) Same (Tamathul)
ii) One being a multiple of the other (Tadakhul)
iii) Parallel – neither (i) nor (ii) and have no common divisor (Tabayin)
iv) Converge – neither (i) nor (ii) but have a common divisor (Tawafuq)

The reader is encouraged to take some time and arbitrarily white down any two numbers then consider the relationship between them. It MUST NECESSARILY be one of these four!

Rule G: When there is a common divisor between the denominators of the shares, divide one by the common divisor and multiply the solution with the other. The result is the base number.

Example 12

HeirsHusbandGrandmotherSon
Shares¼1/6Resiue
Base number12
Portions327

Common divisor of 4 and 6 = 2
LCM of 4 and 6 = 4 ÷ 2 = 2 × 6 = 12   or   6 ÷ 2 = 3 × 4 = 12
Husband: 12 × ¼ = 3 portions
Grandmother: 12 × 1/6 = 2 portions
Son: 12 – 3 – 2 = 7 portions   or   12 – (3 + 2) = 7 portions


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