CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 295
Divisibility
There are five different ways of saying that one
integer, a, is a multiple of another integer, b.
Understand each phrasing.- ais divisible by b.
- ais a multiple of b.
- bis a factor (divisor) of a.
- When ais divided by b, the remainder is 0.
- a/bis an integer.
Example:
42 is a multiple of 7, so- 42 is divisible by 7.
- 42 is a multiple of 7.
- 7 is a factor (divisor) of 42.
- When 42 is divided by 7, the remainder is 0.
- 42/7 is an integer (6).
To see if integer ais divisible by integer b, divide
aby bon your calculator and see whether the
result is an integer. Or use one of the quick
checks below.- Multiples of 3 have digits that add up to a mul-
tiple of 3.
Example:
345 is a multiple of 3 because 3 + 4 + 5 =12, which
is a multiple of 3.- Multiples of 5 end in 0 or 5.
- Multiples of 6 end in an even digit, and their
digits add up to a multiple of 3. - Multiples of 9 have digits that add up to a mul-
tiple of 9.
Example:
882 is a multiple of 9 because 882 ÷ 9 =98, which
is an integer, and because its digit sum
(8 + 8 + 2 =18) is a multiple of 9.- If an integer is a multiple of 10, it ends in 0.
Remainders
A remainder is a whole number “left over”
when one whole number is divided by another
whole number a whole number of times.Think about giving balloons to kids: you can
only have a whole number of balloons, a whole
number of kids, and you can’t give any kid a
fraction of a balloon. If you try to divide 34 bal-
loons among 4 kids, each kid can get 8 bal-
loons, but then you will have 2 balloons “left
over.” This is your remainder.To find a remainder with a calculator, divide
the two whole numbers on your calculator,
then multiply the “decimal part” of the result
by the divisor.Example:
What is the remainder when 34 is divided by 5?
34 ÷ 5 =6.8 and .8 × 5 = 4Remainders can be very useful in solving SAT
“pattern” problems.Example:
What is the 50th term in this sequence? 7, 9, 3, 1,
7, 9, 3, 1,...
The pattern repeats every four terms. The remain-
der when 50 is divided by 4 is 2, so the 50th term
is the same as the 2nd term, which is 9.Primes, Evens, and Odds
•A prime numberis any integer greater than 1 that
is divisible only by 1 and itself (like 2, 3, 5, 7, 11,
13, 17,.. .).- An even numberis any multiple of 2. It can al-
ways be expressed as 2n,where nis some integer.
e.g., 28 =2(14) - An odd numberis any integer that is nota multi-
ple of 2. Any odd number can be expressed as
2 n+1, where nis some integer. e.g., 37 =2(18) +1.
Be careful not to confuse oddwith negative(and
evenwith positive). Students commonly do this
because oddand negativeare “bad” words and
evenand positiveare “good” words. To avoid
this mistake, pay special attention to the words
odd, even, negative, and positiveby underlining
them when you see them in problems.Lesson 7: Divisibility