Divisibility and Remainders
A number (
x) is divisible
by another number (
y) if, when
x is
divided by
y, the answer is a whole
number. For example,
6 is divisible by
3 because
,
and
2 is a whole number.
6 is not divisible
by
4, because
= 1 2
/4 = 1
1/2,
which is not a whole number. Another way of describing
is
to say that you can make one complete division with a remainder
of
2.
The SAT will sometimes test whether you can determine
if one number is divisible by another. To check divisibility, you
can always do the division by hand and see if the result is a whole
number. However, if the number you have to divide is large, working
out the problem by hand can be quite difficult. There are some divisibility
rules that make this task much easier—these rules allow us to determine
whether a number is divisible by another number without having to
carry out the division.
Divisibility Rules
-
All whole numbers are divisible by 1.
- All
numbers with a ones digit of 0, 2, 4, 6, or 8 are
divisible by 2.
- A
number is divisible by 3 if its digits add up to a
number divisible by 3. For example, 6,711 is
divisible by 3 because 6 + 7 + 1 + 1 = 15,
and 15 is divisible by 3.
- A
number is divisible by 4 if its last two digits are
divisible by 4. For example, 780, 52, and 80,744 are
divisible by 4, but 7,850 is not.
- A
number is divisible by 5 if it ends in 0 or 5.
- A
number is divisible by 6 if it is even and also divisible
by 3.
- Sorry.
There are no rules for 7.
- A
number is divisible by 8 if its last three digits are
divisible by 8. For example, 905,256 is
divisible by 8 because 256 is divisible
by 8. 74,513 is not divisible by 8 because 513 is
not divisible by 8.
- A
number is divisible by 9 if its digits add up to a
number divisible by 9. For example, 1,458 is
divisible by 9 because 1 + 4 + 5 + 8 = 18 and 18 is
divisible by 9.
- A
number is divisible by 10 if it ends in 0.
Two Notes:
(1) Because a number divided by itself always
yields 1, a number is always divisible by itself. For
example, 7 is divisible by 7, and 8,374 is
divisible by 8,374. (2) No number is divisible
by a number greater than itself.
