SEVERABILITY CRITERIA: WHAT THEY ARE, WHAT THEY ARE FOR AND RULES - MATHS - 2025

The divisibility criteria are theoretical arguments used to determine if a whole number is divisible by another whole number. Since the divisions must be exact, this criterion applies only to the set of integers Z. For example, the figure 123 is divisible by three, according to the divisibility criteria of 3, which will be specified later.

A division is said to be exact if its remainder is equal to zero, the remainder being the differential value obtained in the traditional manual division method. If the remainder is different from zero, the division is inaccurate, and it is necessary to express the resulting figure with decimal values.

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What are the divisibility criteria for?

Its greatest utility is established prior to a traditional manual division, where it is necessary to know if an integer number will be obtained after performing said division.

They are common in obtaining roots by the Ruffini method and other procedures related to factoring. This is a popular tool for students who, for pedagogical reasons, are not yet allowed to use calculators or digital calculation tools.

Most common rules

There are divisibility criteria for many whole numbers, which are mostly used for working with prime numbers. However, they can also be applied with other types of numbers. Some of these criteria are defined below.

Criterion of divisibility of one "1"

There is no specific divisibility criterion for number one. It is only necessary to establish that every integer is divisible by one. This is because every number multiplied by one remains unchanged.

Criterion of divisibility of the two "2"

It is affirmed that a number is divisible by two if its last digit or number referring to the units, is zero or even.

The following examples are observed:

234: It is divisible by 2 because it ends in 4, which is an even figure.

2035: It is not divisible by 2 since 5 is not even.

1200: It is divisible by 2 because its last digit is zero.

Criterion of divisibility of three "3"

A digit will be divisible by three if the sum of its separate digits is equal to a multiple of three.

123: It is divisible by three, since the sum of its terms 1 + 2 + 3 = 6 = 3 x 2

451: It is not divisible by 3, which is verified by verifying that 4 + 5 +1 = 10, it is not a multiple of three.

Criterion of divisibility of four "4"

To determine if a number is a multiple of four, you need to verify that its last two digits are 00 or a multiple of four.

3822: Observing its last two digits "22" it is detailed that they are not a multiple of four, therefore the figure is not divisible by 4.

644: We know that 44 = 4 x 11, so 644 is divisible by four.

3200: As its last figures are 00, it is concluded that the figure is divisible by four.

Divisibility criterion of five "5"

It is quite intuitive that the divisibility criterion of five is that its last digit is equal to five or zero. Since in the table of five it is observed that all the results end with one of these two numbers.

350, 155 and 1605 are according to this criterion figures divisible by five.

Criterion of divisibility of six "6"

For a number to be divisible by six, it must be true that it is divisible at the same time between 2 and 3. This makes sense, because the decomposition of 6 is equal to 2 × 3.

To check divisibility by six, the criteria for 2 and 3 are analyzed separately.

468: By ending in an even number, it meets the divisibility criterion by 2. By separately adding the digits that make up the figure, we obtain 4 + 6 + 8 = 18 = 3 x 6. The divisibility criterion of 3 is met. Therefore, 468 is divisible by six.

622: Its even number corresponding to the units indicates that it is divisible by 2. But when adding its digits separately 6 + 2 + 2 = 10, which is not a multiple of 3. In this way it is verified that 622 is not divisible by six.

Criterion of divisibility of seven "7"

For this criterion, the complete number must be separated into 2 parts; units and remainder of the number. The criterion for divisibility by seven will be that the subtraction between the number without the units and twice the units is equal to zero or a multiple of seven.

This is best understood by examples.

133: The number without the ones is 13 and twice the ones is 3 × 2 = 6. In this way we proceed to carry out the subtraction. 13 - 6 = 7 = 7 × 1. This ensures that 133 is divisible by 7.

8435: Subtraction of 843 - 10 = 833 is performed. Noting that 833 is still too large to determine divisibility, the process is applied once more. 83 - 6 = 77 = 7 x 11. Thus, 8435 is divisible by seven.

Eight "8" divisibility criterion

It must be true that the last three digits of the number are 000 or a multiple of 8.

3456 and 73000 are divisible by eight.

Criterion of divisibility of the nine "9"

Similarly to the divisibility criterion of three, it must be verified that the sum of its separate digits is equal to a multiple of nine.

3438: When the sum is made, we obtain 3 + 4 + 3 + 8 = 18 = 9 x 2. Thus, it is verified that 3438 is divisible by nine.

1451: Adding the digits separately, 1 + 4 + 5 + 1 = 11. As it is not a multiple of nine, it is verified that 1451 is not divisible by nine.

Criterion of divisibility of ten "10"

Only numbers ending in zero will be divisible by ten.

20, 1000, and 2030 are divisible by ten.

Criterion of divisibility of eleven "11"

This is one of the most complex, however working in order guarantees easy verification. For a figure to be divisible by eleven, it must be satisfied that the sum of the digits in even position, minus, the sum of the digits in odd position is equal to zero or a multiple of eleven.

39.369: The sum of the even figures will be 9 + 6 = 15. And the sum of the figures in odd position is 3 + 3 + 9 = 15. In this way, when subtracting 15 - 15 = 0, it is verified that 39,369 is divisible by eleven.

References

1. Criteria for Divisibility. NN Vorobyov. University of Chicago Press, 1980
2. Elementary Number Theory in Nine Chapters. James J. Tattersall. Cambridge University Press, Oct 14 1999
3. History of the Theory of Numbers: Divisibility and primality. Leonard Eugene Dickson. Chelsea Pub. Co., 1971
4. Divisibility by 2-powers of Certain Quadratic Class Numbers. Peter Stevenhagen. University of Amsterdam, Department of Mathematics and Computer Science, 1991
5. Elementary arithmetic. Enzo R. Gentile. General Secretariat of the Organization of American States, Regional Program for Scientific and Technological Development, 1985