Number Theory I Junior Math Circles - University Of Waterloo Faculty Of Mathematics Worksheet With Answers Page 2
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1
2
3
4
5
6
72
Divisibility Rules
The rules listed in the table below help us determine whether a number is divisible
by each of the numbers from 2 to 11. Note that 0 is divisible by all positive numbers.
Divisible by Test
Example
2
The ones digit is even.
46 is divisible by 2 since 6 is even.
3
The sum of all the digits is di-
153 is divisible by 3 since 1+5+3 =
visible by 3.
9, which is divisible by 3.
4
The last two digits are divisi-
820 is divisible by 4 since 20 is di-
ble by 4.
visible by 4.
5
The ones digit is 0 or 5.
795 is divisible by 5 since its last
digit is 5.
6
The number is divisible by 2
258 is divisible by 6 since 8 is even
and 3.
so it is divisible by 2 and 2+5+8 =
15 which is divisible by 3.
7
Subtract twice the ones digit
672 is divisible by 7 since 2
2 = 4
from the rest of the number,
and 67
4 = 63, which is divisible
until you get a small number
by 7.
that is divisible by 7.
8
The last three digits are divis-
4816 is divisible by 8 since 816 is
ible by 8.
divisible by 8.
9
The sum of all the digits is di-
567 is divisible by 9 since 5+6+7 =
visible by 9.
18 which is divisible by 9.
10
The ones digit is 0.
270 is divisible by 10 since the ones
digit is 0.
11
The alternating sum of the
1364 is divisible by 11 since 1-3+6-
digits is divisible by 11.
4=0, which is divisible by 11.
Exercise 2 Using the rules above, check off each of the numbers from 2 to 11
that each number is divisible by.
2
3
4
5
6
7
8
9
10
11
926
1420
7848
22176
39208
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