| Factors and Divisibility |
What is Factoring?
To factor a number, we break it down into other numbers that multiply to give the original.
For example, we can factor 12 as 2 and 6, 3 and 4, 12 and 1.
definition: The factors of a number or expression are the two or more numbers, elements or quantities whose product equals the number or expression.
definition: The prime factors of a given number or expression are all the prime numbers that are factors of the given number or expression.
So the prime factors of 12 are 1, 2, and 3 since 1 × 2 × 2 × 3 = 12. Since 2 × 2 = 2², we could and should write the factors of 12 as 1 × 2²× 3
For example, 6 and 3 or 9 and 2 are factors of 18, but if we want the prime factors, we would have to make 6 = 3 × 2 and we'd make 9 = 3 × 3, so the prime factors of 18 are 3 and 2 since we can express 18 as 3² × 2
We know that 7 and 3 are factors of 21 but they're also prime factors of 21 since both 7 and 3 are prime numbers.
Recall that a prime number is one that can only be expressed as the product of itself and one. So 2, 3, 5, 7, and 11 are all prime numbers since 2 = 2 × 1, 3 = 3 × 1, etc.
Since any even number is divisible by 2, all prime numbers bigger than 2 must be odd numbers.
Factoring is an extremely important mathematical technique since it allows us to write complicated algebric expressions and equations in a simplified product form, which helps us solve equations and inequalities.
The one thing to always remember about factoring is that the expression in factored form must be equal to the original expression. In other words, if you multiply out the factors, you should get exactly what you started with.
We must be fluent with our multiplication tables if we want to factor numbers correctly.
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Divisibility Rules
These rules help us find the factors of given numbers:
A number is divisibe by:
2 if its an even number (ends in 0, 2, 4, 6 or 8) |
3 if the sum of its digits is divisible by 3 |
4 if the last two digits are divisible by 4 |
5 if the number ends in 5 or 0 |
6 if the number is divisible by 2 and by 3 |
8 if the last three digits are divisible by 8 |
9 if the sum of its digits is divisible by 9 |
10 if the number ends in 0 |
examples
| a) 531 is divisible by 3 since 5 + 3 + 1 = 9 | b) 9516 divides by 4 since 16 divides by 4 |
| c) 1128 is divisible by 6 since it is even and the sum of its digits = 12 (divides by 3) |
d) 5024 is divisible by 8 since 024 divides by 8. |
| e) 576 is divisible by 9 since the digit sum = 18 which is divisible by 9. |
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Note: it's a good idea to recognize the squares of the first 12 integers
and the cubes of the first 6 integers.
(an integer is a positive or negative whole number)
| Squares of the first 12 integers | ||||||||||||
| x = | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| x² = | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
| Cubes of the first 6 integers | ||||||
| x = | 1 | 2 | 3 | 4 | 5 | 6 |
| x³ = | 1 | 8 | 27 | 64 | 125 | 216 |
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Practice
1) Express these numbers as a product of 2 factors:
| a) 25 | b) 93 | c) 50 | d) 24 | e) 972 | f) 640 |
2) Which of these numbers are divisible by 3? For those that are, write the factors.
| a) 3654 | b) 786 | c) 9009 | d) 1836 | e) 972 | f) 640 |
3) Which of these numbers are divisible by 4? For those that are, write the factors.
| a) 348 | b) 786 | c) 6536 | d) 18 374 | e) 972 | f) 640 |
4) Which of these numbers have 9 as a factor? For those that are, write the factors.
| a) 549 | b) 486 | c) 6534 | d) 18 378 | e) 972 | f) 640 |
5) There are 36 students in a class. If the dimensions of the classroom are such that no row can have more than 18 seats and no less than 3 seats, list all possible arrangements of the chairs in rows so that we have the same number of students in each row. (There are 6 possibilities).
6) Write these numbers as a product of PRIME numbers. (use exponents where needed)
| a) 28 | b) 64 | c) 135 | d) 165 | e) 840 |
| f) 256 | g) 1416 | h) 1120 | i) 405 | j) 567 |
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Solutions
1) There are many different solutions. Just check that the product is correct.
| a) 25 = 5 × 5 |
b) 93 = 3 × 31 |
c) 50 = 5 × 10 |
d) 24 = 8 × 3 |
e) 972 = 4 × 243 |
f) 640 = 16 × 40 |
2)
| a) 3654 = 3 × 1218 |
b) 786 = 3 × 262 |
c) 9009 = 3 × 3003 |
d) 1836 = 3 × 612 |
e) 972 = 3 × 324 |
f) is not divisible by 3.
3)
| a) yes 348 = 4 × 87 |
b) 786 no | c) yes 6536 = 4 × 1634 |
d) 18 374 no | e) yes 972 = 4 × 243 |
f) yes 640 = 4 × 160 |
4)
| a) yes 549 = 9 × 61 |
b) yes 486 = 9 × 54 |
c) yes 6534 = 9 × 61 |
d) yes 18 378 = 9 × 2042 |
e) yes 972 = 9 × 108 |
f) 640 no |
5)
| # of Rows | # of Seats per Row |
| 2 | 18 |
| 3 | 12 |
| 4 | 9 |
| 6 | 6 |
| 9 | 4 |
| 12 | 3 |
6)
| a) 28 = 2² × 7 |
b) 64 = 26 |
c) 135 = 3³ × 5 |
d) 165 = 5 × 3 × 11 |
e) 840 = 2³ × 3 × 5 × 7 |
| f) 256 = 28 |
g) 1416 = 2³ × 3 × 59 |
h) 1120 = 25 × 5 × 7 |
i) 405 = 34 × 5 |
j) 567 = 34 × 7 |
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