# Factors and Divisibility Rules

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As we all know, a factor is one of two or more quantities that divides a given quantity without a remainder. For example, 3 and 7 are factors of 21; x and y are factors of xy. Concept of factors is a pre-requisite in studying divisibility rules. It is also important in determining whether a certain number is prime or composite.

To determine if a number is prime or composite, we have to find all factors of the number. If the number has only two factors, 1 and itself, then it is prime. If the number has more than two factors, then it is composite.

However, if we deal with large numbers such as 621 then the procedure above would be time-consuming to find all factors of large numbers. Hence, here comes the divisibility rules, which is one way to find factors of large numbers quickly.

Wikipedia has divisibility rules for numbers 1–20 and beyond 20. It also provides step-by-step examples for divisibility rules for numbers 1–20. In our case, we just limit our discussion on divisibility rules for numbers 1-11, 13, 17 and 19.

DIVISIBILITY RULES

Let's look at some tests for divisibility and examples of each. But before we discuss the divisibility tests, we have to understand first the following definitions and examples from Math Goodies.

• One whole number is divisible by another if, after dividing, the remainder is zero.
• Example: 36 is divisible by 9 since 36 ÷ 9 = 4 with a remainder of 0.

• If one whole number is divisible by another number, then the second number is a factor of the first number.
• Example: Since 36 is divisible by 9, 9 is a factor of 36.

• A divisibility test is a rule for determining whether one whole number is divisible by another. It is a quick way to find factors of large numbers.
• Example: Divisibility Test for 3: if the sum of the digits of a number is divisible by 3, then the number is divisible by 3.

Below are some of the divisibility rules. You can view more divisibility rules by visiting Wikipedia.

1. All integers are divisible by 1.
2. All even numbers are divisible by 2. In other words, if the last digit of an integer ends in any of 0, 2, 4, 6, or 8, then the said integer is divisible by 2.
3. Example: 5,672: 2 is even.

4. A number is divisible by 3 if the sum of the digits is a multiple of 3.
5. Example: 1,351,245 is divisible by 3 since 1 + 3 + 5 + 1 + 2 + 4 + 5 equals 21 is clearly divisible by 3.

6. A number is divisible by 4 if the number formed by the tens and ones digits is a multiple of 4 or if the the last two digits are both zeros.
7. Example: 200 is divisible by 4 because the last two digits are zeros; 524 is divisible by 4 because the tens and ones digits form the number 24, a multiple of 4.

8. All numbers that end in 0 or 5 are divisible by 5 or if the ones digit ends in 0 or 5, then the integer is divisible by 5.
9. Example: 2,120 is divisible by 5 because the ones digit is 0.

10. A number that is divisible by both 2 and 3 is divisible by 6.
11. Example: The integer 4,662 is even, so it is divisible by 2. The sum of the digits is 18, a multiple of 3. Hence, 4,662 is divisible by 6.

12. A number is divisible by 7 if the difference between the twice the units digit and the number formed by the remaining digits is a multiple of 7.
13. Example: We will examine if 4,627 is divisible by 7. Using the rule, twice the units digit is $2\times 7 = 14$. Subtracting 14 from the remaining digits (462) gives $462-14=448$, still a big number. Repeat the process; that is, twice the units digit is $2\times 8=16$. Subtracting 16 from 44 gives 28, a multiple of 7. Thus, 4,627 is divisible by 7.

14. (i.a) If the hundreds digit is even, examine the number formed by the last two digits. If it is a multiple of 8, then the given integer is divisible by 8.
Example: The hundreds digit of 1,624, 6, is even. The last two digits, 24, is a multiple of 8. Hence, 1,624 is divisible by 8.
(i.b) If the hundreds digit is odd, examine the number obtained by the last two digits plus 4. If the result is a multiple of 8, then the given integer is divisible by 8.
Example: The hundreds digit of 14,352 is odd. The last two digits plus 4 equals $52+4=56$ is a multiple of 8. Therefor, 14,352 is divisible by 8.
(ii) Add the last digit to twice the rest. If the result is clearly a multiple of 8, then the said integer is divisible by 8.
Example: 352 is divisible by 8 because $(35\times 2)+2=72$, a multiple of 8.
(iii) Add four times the hundreds digit to twice the tens digit to the ones digit. If the result is a multiple of 8, then that integer is divisible by 8.
Example: 34,152 is divisible by 8 because $4\times 1+5\times 2+2=16$, a multiple of 8. Hence, 34,152 is divisible by 8.
15. A number is divisible by 9 if the sum of the digits is a multiple of 9.
16. Example: The sum of the digits of 1,452,725,352 is 36, a multiple of 9. Hence, 1,452,725,352 is divisible by 9.

17. All numbers that end in 0 are divisible by 10.
18. Example: 123,897,467,350 is divisible by 10 because it ends in 0.

19. To determine if a number is divisible by 11, find the difference of the sums of the two groups of alternate digits. If the difference is a multiple of 11, then the original number is also a multiple of 11.
20. Example: The alternate digits of 293,469 in set form are $\left \{2,3,6 \right \}$ and $\left \{9,4,9 \right \}$. The sums of the elements of the two sets are 11 and 22, respectively. Their difference, 22-11, is 11, a multiple of 11. Hence, 293,469 is divisible by 11.

21. To determine of an integer is divisible by 13, form the alternating sum of blocks of three from right to left. If the result is a negative one, get the absolute value of it. Then, add 4 times the last digit to the rest. If the result is a multiple of 13, then it is divisible by 13.
22. Example: Is 8,733,816 divisible by 13? Using the rule, 8,733,816: $\left |-8+733-816\right |=91$. Then for 91, we get $9+1\times 4=13$, a multiple of 13. Hence, 8,733,816 is divisible by 13.

23. To determine if an integer is divisible by 17, subtract 5 times the last digit from the rest.
24. Example: The number 221 is divisible by 17 because $22-1\times 5=17$.

25. For divisibility rule of 19, add twice the last digit to the rest.
26. Example: The integer 437 is divisible by 19 because $43+7\times 2=57$.