Download UNIT 3: DIVISIBILITY 1. Prime numbers

Natalia Luque Sánchez
1º ESO
IES Benalmádena
UNIT 3: DIVISIBILITY
1. Prime numbers
A prime number (or a prime) is a natural number which has exactly two distinct natural
number divisors: 1 and itself.
Prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,...
The number 1 is by definition not a prime number.
2. Factoring
If a number can be expressed as a product of two or more whole numbers, then the whole
numbers are called factors of that number.
6 = 6 x 1 or 6 = 3 x 2
So, the factors of 6 are 1, 2, 3 and 6.
A factor is any number that will divide into another number exactly (with no part left over)
e.g. 8 can be divided by 2 (the factor in this example) 4 times. However, in total the number 8 has
several factors: 1, 2, 4 and 8.
Example:
Factors of 24: 1, 2, 3, 4, 6 ,8, 12, 24
Factors of 27: 1, 3, 9, 27
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
Factors of 61: 1, 61
Numbers that are greater than 1 and has only two positive factors; 1 and itself are called prime
numbers
3. Multiples
The multiple of a number is the product generated when that number is multiplied by an
integer.
The first multiples of a number are obtained by multiplying the number by each of the
natural numbers: 1, 2, 3, 4, 5, ...
So, the multiples of 7 are:
7·1 = 7
7·2 = 14
7·3 = 21
and so on.
Natalia Luque Sánchez
1º ESO
IES Benalmádena
4. Tests of divisibility
Here are some quick and easy checks to see if one number will divide exactly.
Divisible by 2.
A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8.
Example: 2346 is divisible by 2 since the last digit is 6.
Divisible by 3.
A number is divisible by 3 if the sum of the digits is divisible by 3.
Example: 23457 is divisible by 3 since the sum of the digits is 21 (2 + 3 + 4 + 5+ 7 = 21), and 21 is
divisible by 3.
Divisible by 5.
A number is divisible by 5 if the last digit is either 0 or 5.
Example: 12457896535 is divisible by 5 since the last digit is 5.
5. Prime factorization
We can break up a number into its prime numbers. This process is called prime
factorization.
To write 60 in prime factored form we would get: 60 = 2 x 2 x 2 x 3 x 5.
This can be further simplified using exponents to 60 = 23 x 3 x 5.
Prime factoring is to factor and then continue factoring a number until you can no longer
reduce the factors into constituent factors any further.
Any number can be written as a product of prime numbers in a unique way (except for the
order).
To determine the prime factors of a given number:
1. Choose the smallest prime that will divide evenly into the given number and
divide.
2. Repeat step 1 on the answer until the answer is prime.
Find the prime factorization of 24:
24 can be divided by 2 (the smallest prime number). Let's check: 24 ÷ 2 = 12
But 12 is not a prime number, so we need to factor it further: 12 ÷ 2 = 6
But 6 is not a prime number, so we need to factor it further: 6 ÷ 2 = 3
And 3 is a prime number, so:
Natalia Luque Sánchez
1º ESO
IES Benalmádena
24 = 2 x 2 x 2 x 3 is the prime factorization of 24.
It can also be written as 24 = 23 x 3
6. Greatest common factor (GCF)
To find the GCF of a set of numbers, you must factor each of the numbers into primes. Then
for each different prime number in all of the factorizations, do the following...
1. Count the number of times each prime number appears in all the factorizations.
2. For each prime number, take the lowest of these counts and write the result.
3. The greatest common factor is the product of all the prime numbers written down.
Example: GCF (4,6)=2, because 4=2·2 and 6=2·3, so GCF(4,6)=2
If GCF(a,b)=1, is said that a and b are relative primes
example:
Find GCF(72,90,120)
1. Determine the prime factorization of each number:
72=23·32
90=2·32·5
120=23·3·5
2. Take the prime numbers that appears in all the factorizations. (Remember taking the lowest
number of times they appear)
Prime numbers selected: 2 y 3
3. GCF(72,90,120)=2·3=6
7. Lowest common multiple (LCM)
To find the LCM of a set of numbers, you must factor each of the numbers into primes. Then
for each different prime number in all of the factorizations, do the following...
1. Count the number of times each prime number appears in each of the factorizations.
2. For each prime number, take the largest of these counts and write the result.
3. The least common multiple is the product of all the prime numbers written down.
Natalia Luque Sánchez
1º ESO
IES Benalmádena
Example: LCM (4,6)=12, because 4=2·2 and 6=2·3, so LCM(4,6)=2·2·3
example:
Find LCM(16,24,40)
1. Determine the prime factorization of each number:
16=24
24=23·3
40=23·5
2. Take the prime numbers that appears in all the factorizations. (Remember taking the highest
number of times they appear)
3. LCM(16,24,40) = 24·3·5 = 240