# How to Conduct Divisibility Tests

When one number is *divisible* by a second number, you can divide the first number by the second without having anything left over. For example, 16 is divisible by 8 because 16 / 8 = 2, with no remainder. You can use a bunch of tricks for testing divisibility without actually doing the division.

The most common tests are for divisibility by 2, 3, 5, and 11. Testing for divisibility by 2 and 5 are child's play; testing for divisibility by 3 and 11 requires a little more work. Here are some quick tests:

**By 2:**Any number that ends in an even number (2, 4, 6, 8, or 0) is even — that is, it's divisible by 2. All numbers ending in an odd number (1, 3, 5, 7, or 9) are odd — that is, they aren't divisible by 2.**By 3:**Any number whose digital root is 3, 6, or 9 is divisible by 3; all other numbers (except 0) aren't. To find the*digital root*of a number, just add up the digits. If the result has more than one digit, add up*those*digits and repeat until your result has one digit.**By 5:**Any number that ends in 5 or 0 is divisible by 5; all other numbers aren't.**By 11:**Alternately place plus signs and minus signs in front of all the digits and find the answer. If the result is 0 or any number that's divisible by 11 (even if this result is a negative number), the number is divisible by 11; otherwise, it isn't.Always put a plus sign in front of the*Remember:**first number.*

## Sample questions

Which of the following numbers are divisible by 3?

**a.**31**b.**54**c.**768**d.**2,809Add the digits to determine each number’s digital root; if the digital root is 3, 6, or 9, the number is divisible by 3:

**a.**31:**No,**because 3 + 1 = 4 (check: 31 / 3 = 10 r 1)**b.**54:**Yes,**because 5 + 4 = 9 (check: 54 / 3 = 18)**c.**768:**Yes,**because 7 + 6 + 8 = 21 and 2 + 1 = 3 (check: 768 / 3 = 256)**d.**2,809:**No,**because 2 + 8 + 0 + 9 = 19, 1 + 9 = 10, and 1 + 0 = 1 (check: 2,809 / 3 = 936 r 1)Which of the following numbers are divisible by 11?

**a.**71**b.**154**c.**528**d.**28,094Place + and – signs between the numbers and determine whether the result is 0 or a multiple of 11:

**a.**71:**No,**because +7 – 1 = 6 (check: 71 / 11 = 6 r 5)**b.**154:**Yes,**because +1 – 5 + 4 = 0 (check: 154 / 11 = 14)**c.**528:**Yes,**because +5 – 2 + 8 = 11 (check: 528 / 11 = 48)**d.**28,094:**Yes,**because +2 – 8 + 0 – 9 + 4 = –11 (check: 28,094 / 11 = 2,554)

## Practice questions

Which of the following numbers are divisible by 2?

**a.**37**b.**82**c.**111**d.**75,316Which of the following numbers are divisible by 5?

**a.**75**b.**103**c.**230**d.**9,995Which of the following numbers are divisible by 3?

**a.**81**b.**304**c.**986**d.**4,444,444Which of the following numbers are divisible by 11?

**a.**42**b.**187**c.**726**d.**1,969

Following are the answers to the practice questions:

b and d

**a.**37:**No,**because it’s odd (check: 37 / 2 = 18 r 1)**b.**82:**Yes,**because it’s even (check: 82 / 2 = 41)**c.**111:**No,**because it’s odd (check: 111 / 2 = 55 r 1)**d.**75,316:**Yes,**because it’s even (check: 75,316 / 2 = 37,658)a, c, and d

**a.**75:**Yes,**because it ends in 5 (check: 75 / 5 = 25)**b.**103:**No,**because it ends in 3, not 0 or 5 (check: 103 / 5 = 20 r 3)**c.**230:**Yes,**because it ends in 0 (check: 230 / 5 = 46)**d.**9,995:**Yes,**because it ends in 5 (check: 9,995 / 5 = 1,999)a

**a.**81:**Yes,**because 8 + 1 = 9 (check: 81 / 3 = 27)**b.**304:**No,**because 3 + 0 + 4 = 7 (check: 304 / 3 = 101 r 1)**c.**986:**No,**because 9 + 8 + 6 = 23 and 2 + 3 = 5 (check: 986 / 3 = 328 r 2)**d.**4,444,444:**No,**because 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28, 2 + 8 = 10, and 1 + 0 = 1 (check: 4,444,444 / 3 = 1,481,481 r 1)b, c, and d. Note: Answers add up to 0 or a multiple of 11 for numbers divisible by 11:

**a.**42:**No,**because +4 – 2 = 2 (check: 42 / 11 = 3 r 9)**b.**187:**Yes,**because +1 – 8 + 7 = 0 (check: 187 / 11 = 17)**c.**726:**Yes,**because +7 – 2 + 6 = 11 (check: 726 / 11 = 66)**d.**1,969:**Yes,**because +1 – 9 + 6 – 9 = –11 (check: 1,969 / 11 = 179)