Divisibility by 3, 4 | Concept explained

Divisibility by 3

A number can be checked for divisibility by 3 using the following 3 steps:

1) Add up all the digits in the number.
2) Check whether the sum of all the diigts can be divided by 3
3) If the answer is Yes, then the number can be divided by 3.

Concept of the Divisibility by 3 Rule:

Let the number be represented by ABC (with "A", "B" and "C" being the digit of the number).

Therefore ABC = A(100) + B(10) + C.

We can also write the above maths expression of ABC as
ABC = A(99+1) + B(9+1) + C which leads us to
ABC = 99A+A + 9B+ B + C.

Rearranging, ABC = 99A + 9B + A + B + C.

Look at the last 3 terms. It is the addition of the individual digits "A", "B" and "C".
Look also at the leftmost 2 terms. 99A and 9B are definitely divisible by 3 since 9 = 3 x 3 and 99 = 11 x 3.

From the above observations, to know whether number ABC can be divided by 3, we only need to divide the summation of the digits (A + B+ C) by 3 to decide the outcome.

Divisibility by 4

A number can be checked for divisibility by 4 using the following steps:

1) Look at the last 2 digits (least significant) of the number,
2) Check to see if the number formed (last 2 digits) can be divided by 4, and
3) If the answer is Yes, then the original number can be divided by 4.

Concept of the Divisibility by 4 Rule:

Let the number be DEF.
And DEF = D(100) + EF.

The rule states to look at the digits EF only. How about the digit D?

D is the hundred unit. Therefore it has to be 100, 200, 300, ......

100, 200, 300 ... is definitely divisible by 4. So it does not become a concern or a factor for consideration in the divisibility rule. Therefore only the last 2 digits (EF) decides the outcome.

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Divisibility by 8 | Concept Explained

The rule for divisibility is carried out by:

- Look at the last 3 digits (right-most),
- Check if they are divisible by 8,
- If yes, then the original number is divisible by 8.

This concept is similar to that of the divisibility by 4.

Concept:

Taking 1 four-digit number abcd. Re-writting it ==> 1000a + 100b + 10c + d

We see that the 1000a digit is of no concern to the divisibility test as it is definitely divisible by 8 since 1000 / 8 = 125. Even if it is 2000, 30000,...it is still divisible.

This leave us with the only the last 3 digits to decide whether the original number is divisible by 8. This concept is surely less complex than the Divisibility by 7, right?

For more information about the concept for Divisibility by 7, please click here.

To view concept for Divisibility by 9, please click here.

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Divisibility by 7 | Concept Explained

Divisibility rule is applied when the number is may be too big for computation using a calculator. But it is a good exercise to know this rule as it analyze the in-depth "story" of the integration between division and other mathematical operations. Each divisibility tells a different concept or "story".

Divisibility Rule for 7:

• Take the last (rightmost) digit,


• double it,


• Subtract it from the rest (or vice versa),


• check whether the result is divisible by 7,


• if yes, the original is divisible by 7. (This may have to be done a few rounds)

Why it works? The concept is explained below.

Let the number be "ab". It can be re-written as 10a + b.

We doubled the last digit"b" ==> 2b. For what?

Let's deviate a bit.

If a number is divisible by 7, doubling it will also be divisible by 7.

Example: 21 /7 = 3, doubling 21 ==> 42 / 7 = 6 (still divisible after 21 is doubled)

Therefore 10a + b ==> doubling it ==> becomes 20a + 2b.

By the rule stated at the top, we need to subtract "2b" from "a" (the rest of the digit besides the first digit), that is, a - 2b, for divisibility checking.

This means the 20a + 2b need to be negated and to be added to 21a to achieve the targetted a - 2b.

Result become, -(20a + 2b) + 21a ==> -20a -2b + 21a ==> a -2b.

Note now, the added 21a is definitely divisible by 7, therefore, leaving -(20a + 2b) as the deciding factor for divisibility consideration.

If -(20a + 2b) is divisible by 7, then it implies that the a - 2b is also divisible by 7!

Why go through such hassle of doing a - 2b when we can just double the original number, that is, (20a + 2b) of above and get the answer?

The reason is that by doubling the original number, we will get an even larger number that the calculator cannot handle or display. Therefore the rule using a - 2b has to be used since the number of digits is reduced from original. Clear?

:)

For Concept on Divisibility by:
5 and 6 Click here
3 and 4 Click here
2 Click here
11 Click here

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Divisibility by 5, 6 | Concept Explained

Divisibility by 5

The Rule states: Any number ending with a "5" or "0" can always be divided by 5.
(This is so since multiples of "5" end with either a "0"or "5".)

This is one of the simplest rule in the Divisibility checking.

Concept: 5 times any integer always gives another number ending with a "0" or "5". Therefore by just simply looking at the last digit, we can determine whether the original number can be divided by 5.

Divisibility by 6

Rule: If any integer can be divided by 2 and also 3, then the number can be divided by 6.

Concept:
Since 2 and 3 are prime numbers that multiply to give a 6, these 2 numbers suffice to determine whether the original integer can be divided by 6.

For Concept on Divisibility of 7, click here.

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Divisibility by 9 | Concept Explained

Rule for Divisibility by 9:

- Add up all the digits in the number,
- If answer is divisible by 9, then the original number is divisible by 9.

Concept:

Let's take a 4-digit number, abcd. Re-writting it, 1000a + 100b + 10c + d.

It can also be re-written as 999a + 99b + 9c + a + b + c + d.

Look at the left-most 3 terms (999a, 99b, 9c), they are surely divisible by 9.

Thus, it leaves us with the a + b + c + d to determine whether the original number can be divided by 9. This a + b + c + d is the rule for divisibility by 9.

To view other concepts of number divisibility, click here.

:-)
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Conversion of Number from Decimal Form to Proper Fraction

Number comes in many form. And there are times where we need to convert numbers in decimal form to its fraction form before we can proceed with maths calculation.

One method is suggested here and it involves the use of the Divisibility Rule.

Example: Convert 0.35 to fraction.

0.35 can be converted directly to fraction by changing it to 35 / 100.

Scanning through the Divisibility Rule, we noticed that 35 and 100 can be divided by 5 since their last digits (5 and 0 respectively) satisfy the rule.

Therefore dividing both 35 and 100 by 5, 35 / 100 ==> 7 / 20 (final answer).

The above example shows the usefulness and benefits of the Divisibility rule in number format conversion.

For more ideas about the Divisibility Rule, you may click here.

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Creative Approach To Maths Question

Maths is interesting and exciting. You can also be creative through being able to solve or answer challenging maths questions. The approach to solving the problems decides how effective you can be trained for creativity.

Below an example is explained.

Determine whether the expression 6^2x - 2^2x can be divided by 4.

You can solve this maths challenge through various approaches.

One approach is to substitute the variable "x" in the expression by an ambiguous integer. The number obtained, after the substitution, can be easily determined by applying the divisibility by 4 rule. Click on this link to view the divisibility by 4 rule.

However, this approach is mathematically not elegant. It suffices, though, to meet the question objective.

A better approach that looks more mathematically professional, is the application of factorisation to determine the possibility of divisibility by 4.

This approach requires prior knowledge of the factorisation method to the given maths problem.

Factorisation method to the general expression a ^k - b ^k :

a ^k - b ^k = (a - b)(a^k-1 + a^k-2 b + ... + ab^k-2 + b^k-1)

Therefore by re-writing 6^2x - 2^2x = (6 - 2) ( ........), it becomes very obvious that the given challenge can be divide by 4 since the first written factor of the expression is (6 - 2) = 4!

The message here is that there are various ways to achieve the objective of the maths question. It is a matter of which method is creative or interesting and involves mathematically challenging steps. The approach can be direct and less conceptual or conceptually involved with application of maths theorems. Both are, within their own definition, creative in nature.

A new mindset looking at maths as it is, can change this exciting subject to a life-aspiring adventure with challenges that buils up a person's character.

Wonderful maths .....
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Divisibility Rules - Interesting Maths Concept

Everyone has for some instances want to know if an integer can be divided by another (without giving a remainder). Is there a way? Yes, there is.

Listed below are some interesting methods to check whether integer can be divided by another integer. These wonderful technique is called "Divisibility Rules". They are a set of checking rules to quickly identify divisibility.

Here we go .......

Dividing by Two
Look at the last digit. Is it even or odd? If it is odd, the integer cannot be divided by two.

Dividing by Three
Sum up all the digits within the integer. If the sum is divisible by 3, the original integer can then be divided by 3.

Dividing by Four
Look at the last 2 digits and see if they can be divided by 4. If yes, then the original integer can be divided by 4.

Dividing by Five
If the last digit of the integer is 0 or 5, the original integer can be divided by 5.

Dividing by Six
If the integer can be divided by BOTH 2 and 3, then the original number can be divided by 6.

Dividing by Seven
Double (times two) the last digit of the integer and subtract the doubled number from the leftover (original taking away the last digit) in the original. Check whether the result can be divided by 7. If unsure, you can reapply the "divide by 7" rule to the new result obtained to check whether the original integer can be divided by 7.

Dividing by Eight
Check whether the last 3 digits of the integer can be divided by 8. If it can, the original integer can then be divided by 8.

Dividing by Nine
Sum up all the digits within the integer. If the final sum can be divided by 9, then the original integer can be divided by 9.

Dividing by Ten
If the last digit of the integer is 0, then the integer can be divided by 10. This is easy!

With all the rules above, you can impress anyone with your dividing skill.
But note of caution..... YOU WILL BE BUSY entertaining questions!

:-)

Divisibility by 2

We may come across a number and wonder if it can be divided by 2.

Checking concept:
If a number is even, then it can be divided by 2 as "2" is an even number.

For a review of the mathematical relation between odd and even number, you may visit this link for information.

If the number is odd, a remainder will occur and thus not divisible by 2.
This is the simplest among the Divisibility Rule.

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Can It Be Divided By 11?

To be able to decide whether a number can be divided by 11, we need to understand this multiply by 11 concept first.

Click here to review the x 11 mental concept.

It is simply to do the reverse process of the mental multiplication by 11.

Example: Can the number 352 be divided by 11?

Looking at the centre digit, we can deduce that it is the addition of the 2 extreme digits.
As such the number 352 can be divided by 11.

Example: How about 3751 ?

Again, looking at the digit 3 and 7, we can see that their subtraction produces 7-3 = 4.
For the other 2 digits of 5 and 1, their subtraction also results in 5- 1 = 4.
This means that the number 3751 can be divided by 11 (based on the mental multiplication by 11).

Example: How about 542 ?

The centre digit should be 5 - 2 = 3 to be divisible by 11.
Therefore the number 542 can be quickly deduced not able to be divided by 11.

Check other divisibility concept by clicking the link here.

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