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When a number is exactly divided by fixed divisor, the result should be a whole number with a remainder zero. A divisibility test identifies whether a number can be exactly divided by a fixed divisor, without performing the division.
Ways of determining divisibility:
Gain confidence and improved speed by practicing these methods for the fixed divisors, one through twenty.
All integers are divisible by one.
Examples: 0,1,2
0÷1=0
1÷1=1
2÷1=2
Zero, one and two are all divisible by one.
The units digit is either two, four, six, eight or zero.
Example: 204
204÷2=102
204 has units digit 4, and is therefore divisible by 2.
The sum of all the digits is divisible by three.
Example: 45
4+5=9
9÷3=3
Nine is divisible by three, therefore fourty five is divisible by three.
Test 1: The part of the number which is the tens and units digits is divisible by four.
Example: 752
52÷4=13
52 is divisible by 4, therefore 752 is divisible by 4.
Test 2: Double the tens digit and add it to the units digit. The result must be divisible by four.
Example: 752
(2×5)+2=12
12÷4=3
12 is divisible by 4, therefore 752 is divisible by 4.
The units digit is five or zero.
Examples: 100, 105
100÷5=20
105÷5=21
Both 100 and 105 have unit digits 5 or 0, and are therefore both divisible by 5.
The number is divisible by both two and three.
Example: 18
18÷2=9
18÷3=6
Eighteen is divisible by both two and three; it is therefore divisible by six.
Double the units digit and then subtract it from the remaining number.
The result must be divisible by seven.
Example: 385
5×2=10
38-10=28
28÷7=4
28 is divisible by 7, therefore 385 is divisible by 7.
The number made by the three rightmost digits (hundreds, tens, units), is divisible by eight.
Example: 8,000
1,000÷8= 125
1,000 is divisable by 8, therefore 8,000 is divisable by eight.
The sum of all the digits is divisible by nine.
Example: 4,095
4+0+9+5=18
18÷9=2
18 is divisible by 9, therefore 4,095 is divisible by 9.
The last digit of a number is zero.
Example: 1,000
1,000÷10=100
1,000 has units digit 0, and is therefore divisible by 10.
Make separate sums of odd-position digits and even-position digits.
The difference of these sums must be a multiple of eleven.
Example: 9,899,978.
9+9+9+8=35
8+9+7=24
35-24=11
11 is a multiple of 11, therefore 9,899,978 is divisible by eleven.
The number is divisible by both three and four.
Example: 108
108/3=36
108/4=27
108 is divisible by both 3 and 4, and is therefore divisible by 12.
The result of the mathematical expression, (x+4y), must be divisible by thirteen.
y = units digit
x = remaining digits.
Example: 234
(x + 4y)
(23 + 4(4))
23+16=39
39÷13=3
39 is divisible by 13, therefore 234 is divisible by 13.
Even number divisible by seven.
Example: 70
even number
70/7=10
70 is an even number that is divisible by 7, therefore 70 is divisible by 14.
The sum of the digits is divisible by three, and the units digit is zero or five.
Example: 270
units digit: 0
2+7+0=9
9÷3=3
The unit digit is zero, and 9 is divisible by 3, therefore 270 is divisible by 15.
The last four digits (units, tens, hundreds, thousands) of a number is divisible by sixteen.
Example: 21,024
1,024÷16= 64
1,024 is divisible by 16, therefore 21,024 is divisible by 16.
The result of the mathematical expression, (x-5y), must be divisible by seventeen.
y = units digit
x = remaining digits.
Example: 289
(x-5y)
(28-5(9))
28-45=-17
39÷13=3
-17 is divisible by 17, therefore 289 is divisible by 17.
All even numbers that are divisible by nine.
Example: 270
even number
270÷9=30
270 is an even number divisible by 9, therefore it is also divisible by eighteen.
The result of the mathematical expression, (x+2y), must be divisible by nineteen.
y = units digit
x = remaining digits.
Example: 209
(x+2y)
(20+2(9))
20+18=38
38÷19=2
38 is divisible by 19, therefore 209 is divisible by 19.
The number is divisible by both four and five.
Example: 400
400÷4=100
400÷5=80
400 is divisible by both 4 and 5, therefore it is divisible by 20.