Tuesday, 5 August 2008

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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