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Math Review Study Guide
Divisibility Rules
A number is divisible by another number if it can be divided equally by that number; that is, if it yields a whole
number when divided by that number. For example, 6 is divisible by 3 (we say "3 divides 6") because 6/3 = 2 , and 2
is a whole number. 6 is not divisible by 4, because 6/4 = 1.5, and 1.5 is not a whole number.
It is often useful to know whether a number divides another number. To check divisibility, one can always do the
division out by hand and see if the result is a whole number. However, if the number we are dividing is large, this
becomes very difficult. There are some divisibility rules which 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 by 1
A number does not change when it is divided by 1. Therefore, every whole number is divisible by 1.
Divisibility by 2, 4, and 8
All even numbers are divisible by 2. Therefore, a number is divisible by 2 if it has a 0, 2, 4, 6, or 8 in the ones place.
For example, 54 and 2,870 are divisible by 2, but 2,221 is not divisible by 2.
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 divisible by 4. To check whether a number is divisible by 4, just divide the last two digits of the
number by 4. If the result is a whole number, then the original number is divisible by 4.
A number is divisible by 8 if its last three digits are divisible by 8. For example, 880 and 905,256 are divisible by 8
but 74,513 is not divisible by 8. To check divisibility by 8, divide the last three digits of the number by 8. If the result
is a whole number, then the original number is divisible by 8.
Divisibility by 3, 6, and 9
A number is divisible by 3 if its digits add up to a number divisible by 3. For example, 671,451 is divisible by 3
because 6 + 7 + 1 + 4 + 5 + 1 = 24, and 24 is divisible by 3. 84,950 is not divisible by 3, because 8 + 4 + 9 + 5 + 0 =
26, and 26 is not divisible by 3.
A number is divisible by 6 if it is divisible by 2 and 3. In other words, a number is divisible by 6 if its digits add up to a
number divisible by 3 and it ends with an even number. 671,451 (our previous example) is divisible by 3, but not by
2, because 1 is not an even number. Therefore, 671,451 is not divisible by 6. On the other hand, 789,312 is divisible
by 6, because 7 + 8 + 9 + 3 + 1 + 2 = 30, which is divisible by 3, and its last digit is 2, which is even.
A number is divisible by 9 if its digits add up to a number divisible by 9. For example, 91,458 is divisible by 9
because 9 + 1 + 4 + 5 + 8 = 27 and 27 is divisible by 9. 18,453 is not divisible by 9 because 1 + 8 + 4 + 5 + 3 =
21 and 21 is not divisible by 9.
Divisibility by 5
A number is divisible by 5 if it ends in 0 or 5. For example, 875 and 600 are divisible by 5, but 76,508 is not divisible
by 5.
Divisibility by 10
A number is divisible by 10 if it ends in 0. For example, 65,110 and 90 are divisible by 10, but 6,007 is not divisible
by 10.
Note: 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.
Note: No number is divisible by a number greater than itself.
Factors
A factor of a number divides the number. For example, 2, 3, and 4 are all factors of 12. Similarly, we say that 12 is a
multiple of 2, or a multiple of 3, because one can multiply 2 by a whole number to get 12. A number is a multiple of
its factors. To determine whether a number is a factor of another number, we generally use the divisibility
rules discussed last section.
Prime Number
A prime number is a number divisible by only 1 and itself (the number 1 itself is not considered prime). For example,
17 is prime because it is divisible by only 1 and 17. The first few primes, in increasing order, are 2, 3, 5, 7, 11, 13,
17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ...
To determine whether a number is prime, it is not necessary to check whether it is divisible by every number less
than itself. It is only necessary to check whether it is divisible by any prime up to half its value. One might do the
following computations to check whether 91 is prime:
1.
2.
3.
4.
Is 91 divisible by 2? No, it does not end with an even number.
Is 91 divisible by 3? No, 9 + 1 = 10 , which is not divisible by 3.
Is 91 divisible by 5? No, 91 does not end with 0 or 5.
Is 91 divisible by 7? Yes! 91/7 = 13.
Therefore, 91 is not prime.
Composites
A composite number is a number that has at least one factor besides 1 and itself, or any number which is not prime
(although the number 1 is neither prime nor composite). For example, 91 is a composite number. To determine
whether a number is composite, determine whether or not it is prime. If it is prime, it is not composite; if it is not
prime, then it is composite.
Prime Factorization
It is often useful to write a number in terms of its prime factorization, or as the product of its prime factors. For
example, 56 can be written as 2×2×2×7 and 84 can be written as 2×2×3×7. Every number can be written as a
product of primes, and, like a fingerprint, every number has a unique prime factorization.
To take a prime factorization of a number, start by dividing the number by its lowest prime factor. Write down this
factor, and divide the new number by its lowest prime factor (it does not matter if this is the same as the first prime
factor). Write this factor down and divide the new number by its lowest factor. Continue in this manner until the
resulting number is prime. Write this number down as the final factor.
Example 1: Compute the prime factorization of 1,575.
Step 1. Is 1,575 divisible by 2? No. By 3? Yes. 1, 575/3 = 525 . Write down 3.
Step 2. Is 525 divisible by 3? Yes. 525/3 = 175 . Write down 3.
Step 3. Is 175 divisible by 3? No. By 5? Yes. 175/5 = 35 . Write down 5.
Step 4. Is 35 divisible by 5? Yes. 35/5 = 7 . Write down 5.
Step 5. 7 is prime. Write down 7.
Therefore, the prime factorization of 1,575 is 3×3×5×5×7 .
Example 2. Compute the prime factorization of 23,100.
Step 1. 23, 100/2 = 11, 550 . Write down 2.
Step 2. 11, 550/2 = 5, 775 . Write down 2.
Step 3. 5, 775/3 = 1, 925 . Write down 3.
Step 4. 1, 925/5 = 385 . Write down 5.
Step 5. 385/5 = 77 . Write down 5.
Step 6. 77/7 = 11 . Write down 7.
Step 7. 11 is prime. Write down 11.
Therefore, the prime factorization of 23,100 is 2×2×3×5×5×7×11 .
Greatest Common Factor
A common factor of two numbers is a factor that divides both numbers. The greatest common factor (GCF) of two
numbers is the greatest number that divides both numbers. To find the GCF, take the prime factorization of both
numbers. Then write down the factors that they have in common. If they share more than one of the same factor
(two 2's, for example), write them both down. Then multiply the factors they have in common.
For example, the greatest common factor of 1,575 and 23,100 is3×5×5×7 = 525 . 1,575 and 23,100 are both
divisible by 525, and they are not both divisible by any number greater than 525.
Least Common Multiple (LCM)
The least common multiple, or LCM, of two numbers is the smallest number that is divisible by both numbers. To
find the LCM, take the prime factorization of both numbers. Then make a list of the "minimum" factors required to
obtain both numbers. If the prime factorization of one number contains two 3's and the prime factorization of the
other number contains five 3's, write down five 3's.
For example, the least common multiple of 1,575 and 23,100 is2×2×3×3×5×5×7×11 = 69, 300 . 69,300 is divisible
by both 1,575 and 23,100, and there is no number smaller than 69,300 that is divisible by both.
Another way to find the LCM is to multiply the two numbers and divide by the GCF. For example, 1, 575×23, 100 =
36, 382, 500 . 36, 382, 500/525 = 69, 300 . This method is useful when one has a calculator and has already
calculated the GCF.
If two numbers are relatively prime, their LCM is the same as their product. Using the second method for calculating
the LCM, it is easy to see why this is true. The greatest common factor of two relatively prime numbers is 1, so when
the two numbers are multiplied and the result is divided by 1 (the GCF), the result does not change.
The least common multiple of 21 and 40, since they are relatively prime, is 21×40 = 840 .
MLA Citation:
SparkNotes Editors. “SparkNote on Whole Numbers.” SparkNotes.com. SparkNotes LLC. n.d.. Web. 10 Sept. 2013.
Equivalent Fractions
Two fractions are equivalent if they express the same part of a whole. For example, 2/3 and 4/6 express the same
part of a whole. 12/9 and 4/3 are also equivalent.
Two fractions are equivalent if there is a number by which both the numerator and the denominator of one fraction
can be multiplied or divided to yield the other fraction. For example, we can multiply the numerator and denominator
of 2/3 by 2 to yield 4/6, and we can divide the numerator and denominator of 12/9 by 3 to yield 4/3.
To find a fraction that is equivalent to another fraction but has a specified (different) denominator, determine what
the old denominator must be multiplied by to yield the new denominator. Then multiply the old numerator by that
same number. For example, to find a fraction equivalent to 2/9 with a denominator of 45:
1. 9×5 = 45
2. 2×5 = 10
The fraction equivalent to 2/9 is 10/45.
Reducing Fractions
Some fractions, like 6/8, can be written as other fractions with a lower denominator. 6/8 = 3/4 (Note that 6/8 and 3/4
are equivalent by the above definition). Others, like 5/8, cannot be written with a lower denominator. 3/4 and 5/8 are
said to be in lowest terms because they cannot be reduced further.
How does one know which fractions can be reduced and which cannot be reduced? In fractions that can be reduced
(fractions not in lowest terms), the numerator and the denominator share at least one common factor. In fractions
that cannot be reduced (fractions in lowest terms), the numerator and the denominator share no common factors;
that is, they are relatively prime.
To write a fraction in lowest terms, factor the numerator and the denominator. Then divide both the numerator and
the denominator by the greatest common factor. For instance, take the following steps to factor 36/126:
1. Factor. 36 = 2×2×3×3 and 126 = 2×3×3×7 .
2. Find the GCF. The GCF of 36 and 126 is 2×3×3 = 18 .
3. Divide. 36/18 = 2 and 126/18 = 7 .
The reduced fraction is 2/7.
Least Common Denominator (LCD)
A common denominator of two numbers is a number that can be divided by the denominators of both numbers. For
example, 1/6 and 4/9 have common denominators of 18, 36, 54, 72, etc. The least common denominator, or LCD, is
the lowest number that can be divided by the denominators of both numbers. For example, 18 is the least common
denominator of 1/6 and 4/9.
The least common denominator of two fractions is the least common multiple of their denominators. 18 is the LCM
of 6 and 9.
Uses of the Least Common Denominator
The least common denominator is a helpful tool in allowing you to take two different fractions (ex. 3/4 and 7/11) and
write them as equivalent fractions with the same denominator (ex. 33/44 and 28/44). Such a tool is important in
comparing the size of fractions and because fractions can only be added and subtracted from each other when they
have the same denominator. The first step in the process is to find the LCD. Then write each fraction as an
equivalent fraction with the LCD as a new denominator, using the two steps detailed in the section on equivalent
fractions.
Example 1: Write 3/14 and 4/21 as fractions with the same denominator.
I. Find the LCD
1. Factor the denominators. 14 = 2×7 and 21 = 3×7 .
2. Find the LCM of the denominators. 2×3×7 = 42 -or-14×(21/7) = 42 .
3. The LCD is 42.
II. Write each fraction as an equivalent fraction with the LCD (42) as the new denominator.
(a) 14×3 = 42 . 3×3 = 9 .
(b) 21×2 = 42 . 4×2 = 8 .
Thus, 3/14 = 9/42 and 4/21 = 8/42 .
Note: The number by which the numerator must be multiplied in Part II will be the product of the factors of the other
denominator that are not factors of its denominator. Here, 3 was multiplied by 3, which is a factor of 21 but not of 14,
and 4 was multiplied by 2, which is a factor of 14 but not of 21.
Example 2: Write 2/5, 5/12, and 9/8 as fractions with the same denominator.
I. Find the LCD.
1. Factor the denominators. 5 = 5 , 12 = 2×2×3 , and 8 = 2×2×2 . 2. Find the LCM of the denominators. 2×2×2×3×5
= 120 3. The LCD is 120.
II. Write each fraction as an equivalent fraction with the LCD (120) as the new
denominator.
(a) 5×24 = 120 . 2×24 = 48 .
(b) 12×10 = 120 . 5×10 = 50 .
(c) 8×15 = 120 . 9×15 = 135 .
Thus, 2/5 = 48/120 , 5/12 = 50/120 , and 9/8 = 135/120 .
Determining Which Fraction is Greater
It is very difficult to tell whether 3/14 is greater or less than 4/21 just by looking at the two fractions. Here's where
writing them with a common denominator comes in handy. We know that 3/14 = 9/42 and that 4/21 = 8/42 . Since 9
is greater than 8, 9/42 is greater than 8/42. Thus 3/14 is greater than 4/21.
To determine which of two fractions is greater, write them as fractions with the same denominator. Then see which
new fraction has the larger numerator. This is the greater fraction.
Textbook Pages:
152-153 Divisibility Rules, Prime Number Composite Number
156-157 Factors, Prime Factorization
160-161 Greatest Common Factor (GCF)
172-173 Equivalent Fractions
178-179 Comparing and Ordering Fractions
Adding and Subtracting Fractions
We can only add or subtract fractions when they have the same denominator. Therefore, the first step in adding or
subtracting fractions is writing them as fractions with the same denominator. Once the denominators have been
equalized, adding or subtracting the fractions is easy--simply add or subtract the numerators, while keeping the
denominator the same. The numerator of the answer is this result, and the denominator of the answer is the
common denominator.
It is often useful to write the answer in lowest terms, using the steps learned in Reducing Fractions and the Least
Common Denominator.
Example 1: 1/12 + 5/42 = ?
I. Find the LCD.
1. Factor the denominators. 12 = 2×2×3 and 42 = 2×3×7 2. Find the LCM of the denominators. 2×2×3×7 =
84 3. The LCD is 84.
II. Write each fraction as an equivalent fraction with the LCD (84) as the new denominator.
(a) 12×7 = 84 . 1×7 = 7
(b) 42×2 = 84 . 5×2 = 10
Thus, 1/12 = 7/84 and 5/42 = 10/84
III. Add. 7 + 10 = 17
7/84 + 10/84 = 17/84 .BR>
IV. Reduce. Since 17 and 84 have no common factors, the fraction cannot be reduced further.
1/12 + 5/42 = 17/84
Example 2: 13/20 - 3/70 = ?
I. Find the LCD
1. 20 = 2×2×5 and 70 = 2×5×7
2. 2×2×5×7 = 140
3. The LCD is 140
II. Write as equivalent fractions with the LCD as the denominator.
(a) 20×7 = 140 . 13×7 = 91
(b) 70×2 = 140 . 3×2 = 6
Thus, 13/20 = 91/140 and 3/70 = 6/140
III. Subtract. 91 - 6 = 85
91/140 - 6/140 = 85/140
IV. Reduce.
1. Factor the numerator and the denominator. 85 = 5×17 and140 = 2×2×5×7
2. Find the GCF. The GCF is 5.BR> 3. Divide. 85/5 = 17 and140/5 = 28 . Thus, 85/140 = 17/28
13/20 - 3/70 = 17/28 .
Example 3: 9/8 - 5/12 - 2/5 = ?
I and II. As we have already learned, these three fractions with common denominators are:
9/8 = 135/120
5/12 = 50/120
2/5 = 48/120
III. Subtract. 135 - 50 - 48 = 37
135/120 - 50/120 - 48/120 = 37/120
IV. Reduce. Since 37 and 120 have no common factors, the fraction cannot be reduced further.
9/8 - 5/12 - 2/5 = 37/120
Adding and Subtracting Mixed Numbers
To add and subtract mixed numbers, first add or subtract the whole numbers and then add or subtract the fractions
as above. If the fractional part is improper, convert it to a mixed number.
Example: 5 3/4 + 6 5/6 = ?
5 + 6 = 11
3/4 + 5/6 = ?
I. The LCD of 3/4 and 5/6 is2×2×3 = 12
II. 3/4 = 9/12 and 5/6 = 10/12
III. 9/12 + 10/12 = 19/12.
IV. 19/12 cannot be reduced further
V. As a mixed number, 19/12 = 1 7/12
11 + 1 7/12 = 12 7/12
Thus, 5 3/4 + 6 5/6 = 12 7/12
Multiplying Fractions and Mixed Numbers
When we ask, "What is 4/5 of 55?" or "What is 1/6 of 18/5?", we are really asking, "What is 4/5 times 55?" and
"What is 1/6 times18/5?". When dealing with fractions, the word "of" indicates multiplication. So how does one
multiply fractions?
The first step in multiplying fractions is to change all mixed numbers into improper fractions (SeeConverting Mixed
Fractions. For example, 2 2/3 becomes 8/3. Converting mixed numbers into improper fractions makes them easier
to multiply.
To multiply two (proper or improper) fractions, multiply their numerators together and then multiply their
denominators together--these two numbers will be the numerator and the denominator of the new fraction. For
example,
×
=
because 8×2 = 16 and 3×7 = 21 .
Canceling Out
If the numerator and the denominator have a common factor, we can divide by the common factor to reduce the
fraction to lowest terms and make the multiplication easier to carry out. Since the numerators of the fractions we are
multiplying become a single numerator and the denominators become a single denominator, we can also cancel out
factors of the numerator of one fraction with factors of the denominator of the other. For example,
×
=
×
=
×
=
Note that, in the second step, the "3" in the numerator and the "9" in the denominator
reduced to a "1" in the numerator and a "3" in the denominator.
Dividing Fractions
To divide fractions, we must again change all mixed numbers into improper fractions. Then we note that since
multiplication and division are inverses of each other, multiplying by 4 is the same as dividing by 1/4. Similarly,
dividing by a fraction is the same as multiplying by its inverse. To find the inverse of a fraction, switch the numerator
and the denominator. If the fraction is a whole number, then it can be written as the whole number over 1, and its
inverse is 1 over the whole number. Thus,to divide by a fraction, multiply by its inverse.
For instance, (1 1/3) / (1 3/5) = (4/3)/(8/5) = (4/3)×(5/8) = 20/24 = 5/6
SparkNotes Editors. “SparkNote on Fractions.” SparkNotes.com. SparkNotes LLC. n.d.. Web. 10 Sept. 2013.