Numbers Divisibility Rules

Divisibility rules help us to easily test if one number can be evenly divided by another. But before we continue, let us go ahead explain what is meant by “Divisible by”.

 Divisible By

"Divisible By" means "when you divide one number by another the result is a whole number"

Examples:

14 is divisible by 7, because 14÷7 = 2 exactly

But 15 is not divisible by 7, because 15÷7 = 2 ¹/₇ (i.e., the result is not a whole number)

Note that “Divisible by" and "can be evenly divided by" mean the same thing.

Study the following carefully:

Considering the multiples of 2 = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 ...} we see that the last digit is either zero or divisible by 2.

Considering the multiples of 3 = {3, 6, 9, 12, 15, 18, 21, 24, 27 ...} when we add the digits of any of the multiples, we get a number which is divided by 3, e.g. for 15 we have 1 + 5 = 6, 21 we have 2 + 1 = 3, and for 27 we have 2 + 7 = 9.

Considering the multiples 0f 11 = {11, 22, 33, 44, 55, 66, 77, 88, 99, 121, 132 ...} here, find the difference between the sum of its odd and even digits e.g. sum of odd digits 1 + 1 = 2, sum of even digits = 2 difference = 2 – 2 = 0.  For 132: sum of odd digits = 1 + 2 = 3, even digit = 3 difference = 3 – 3 = 0.

Considering the multiples of 4, 5, 6, 7, 8, 9 and 10, can a general statement involving patterns about these multiples be made?

Yeah, we may discover the following pattern about the multiples of the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 which we called divisibility tests.

The patterns are tabulated as follows:

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