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Divisibility
1.
2. Before we study divisibility, we must
remember the division algorithm.
r
dividend = (divisor ⋅ quotient) + remainder
3. Anumber is divisible by another
number if the remainder is 0 and
quotient is a natural number.
4. If a number is divided by itself then
quotient is 1.
If a number is divided by 1 then quotient
is itself.
If 0 is divided by any none zero number
then quotient is 0.
If any number is divided by zero then
quotient is undefined.
5. Divisibility by 2:
A natural number is divisible by 2 if
it is even, i.e. if its units (last) digit is
0, 2, 4, 6, or 8.
Example: Check if each number is divisible
by 2.
a. 108 b. 466 c. 87 682 d. 68
241
e. 76 543 010
6. Divisibility by 3:
A natural number is divisible by 3 if
the sum of the digits in the number is
multiple of 3.
Example: Determine whether the
following numbers are divisible by 3 or
not.
a) 7605
b) 42 145
c) 555 555 555 555 555
7. Divisibility by 4:
A natural number is divisible by 4 if
the last two digits of the number are
00 or a multiple of 4.
Example: Determine whether the
following numbers are divisible by 4 or
not.
a) 7600
b) 47 116
c) 985674362549093
8. Example: 5m3 is a three-digit number
where m is a digit. If 5m3 is divisible
by 3, find all the possible values of m.
Example: a381b is a five-digit number
where a and b are digits. If a381b is
divisible by 3, find the possible values
of a + b.
9. Example: t is a digit. Find all the possible
values of t if:
a) 187t6 is divisible by 4.
b) 2741t is divisible by 4.
10. Divisibility by 5:
A natural number is divisible by 5 if its
last digit is 0 or 5.
Example: m235m is a five-digit number
where m is a digit. If m235m is divisible
by 5, find all the possible values of m.
11. Divisibility by 6:
A natural number is divisible by 6 if it is
divisible by both 2 and 3.
Example: Determine whether the
following numbers are divisible by 6 or
not.
a) 4608
b) 6 9030
c) 22222222222
12. Example: 235mn is a five-digit number
where m and n are digits. If 235mn is
divisible by 5 and 6, find all the possible
pairs of m, n.
13. Divisibility by 8:
A natural number is divisible by 8 if the
number formed by last three digits is
divisible by 8.
Example: Determine whether the
following number is divisible by 8 or
not.
a) 5 793 128
b) 7265384
c) 456556
14. Divisibility by 9:
A natural number is divisible by 9 if the
sum of the digits of the number is
divisible by 9.
Example: 365m72 is a six-digit number
where m is a digit. If 365m72 is
divisible by 9, find all the possible
values of m.
Example: 5m432n is a six-digit number
where m and n are digits. If 5m432n is
divisible by 9, find all the possible
values of m + n.
15. Divisibility by 10:
A natural number is divisible by 10 if its
units (last) digit is 0.
Example: is 3700 divisible by 10?
16. Divisibility by 11:
A natural number is divisible by 11 if the
difference between the sum of the odd-
numbered digits and the sum of the even-
numbered digits is a multiple of 11.
Example: is 5 764 359 106 divisible by
11?
