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Multiples and Divisibility
A multiple of a number is a product of that number and an integer.
Divisibility:
A number b is said to be divisible by another number a if b is a multiple of a.
Examples:
45 is divisible by 5 because 45 is a multiple of 5 (45 = 9 ⋅ 5)
20 is divisible by 4 because 20 is a multiple of 4 (20 = 5 ⋅ 4)
Saying that b is divisible by a means that b ÷ a results in a remainder of zero. When this
happens, we sometimes say that a divides b “evenly.”
Example:
1. Determine whether 138 is divisible by 4.
Solution:
34
4 138
Since the remainder is not 0, we know that 138 is not divisible by 4.
12
18
16
2
Divisibility Rules
We will examine divisibility rules for 2, 3, 5, 6, 9 and 10.
Divisibility by 2: A number is divisible by 2 (is even) if it has a ones digit of 0, 2, 4, 6, or 8 (that is, it
has an even ones digit).
Examples:
Determine whether each of the following numbers is divisible by 2.
1. 945
2. 3488
Solution:
1. 945 is NOT divisible by 2, because the last digit is odd.
2. 3488 is divisible by 2, because the last digit is even.
3. 3200 is divisible by 2, because the last digit is even.
3. 3200
Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
Examples:
Determine whether each of the following numbers is divisible by 3.
1. 54
2.
480
3. 923
Solution:
1. 54 is divisible by 3, because the sum of the digits is 5 + 4 = 9 is divisible by 3. (9 ÷ 3 = 3)
2. 480 is divisible by 3, because the sum of the digits is 4 + 8 + 0 = 12 is divisible by 3. (12 ÷ 3 = 3)
3. 923 is NOT divisible by 3, because the sum of the digits is 9 + 2 + 3 = 14 is not divisible by 3.
Divisibility by 5: A number is divisible by 5 if its ones digit is 0 or 5.
Examples:
Determine whether each of the following numbers is divisible by 5.
1. 945
2.
3,488
Solution:
3.
3,200
1. 945 is divisible by 5, because the last digit is 5.
2. 3,488 is NOT divisible by 5, because the last digit is not 0 or 5.
3. 3,200 is divisible by 5, because the last digit is 0.
Divisibility by 6: A number is divisible by 6 if its ones digit is 0, 2, 4, 6, or 8 (is even) and the sum of
its digits is divisible by 3.
Examples:
Determine whether each of the following numbers is divisible by 6.
1. 945
2.
3,488
Solution:
3.
4,200
1. 945 is NOT divisible by 6, because the last digit is odd. Thus, it is not divisible by 2.
2. 3,488 is NOT divisible by 6, because the sum of the digits 3 + 4 + 8 + 8 = 23 is not divisible by 3.
Although, it is divisible by 2.
3. 4,200 is divisible by 6, because the last digit is even and the sum of the digits 4 + 2 + 0 + 0 = 6 is
divisible by 3.
Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
Examples:
Determine whether each of the following numbers is divisible by 9.
1. 945
2.
3,488
Solution:
3.
3,200
1. 945 is divisible by 9, because the sum of the digits 9 + 4 + 5 = 18 is divisible by 9.
2. 3,488 is NOT divisible by 9, because the sum of the digits 3 + 4 + 8 + 8 = 23 is not divisible by 9.
3. 3,200 is NOT divisible by 9, because the sum of the digits 3 + 2 + 0 + 0 = 5 is not divisible by 9.
Divisibility by 10: A number is divisible by 10 if its ones digit is 0.
Examples:
Determine whether each of the following numbers is divisible by 10.
1. 945
2.
3,488
3.
Solution:
1. 945 is NOT divisible by 10, because the last digit is not 0.
2. 3,488 is NOT divisible by 10, because the last digit is not 0.
3. 3,200 is divisible by 10, because the last digit is 0.
3,200
Factorizations
Factors and Factorizations:
A number c is a factor of a if a is divisible by c.
A factorization of a expresses a as a product of two or more numbers.
Examples:
1. Find all factors of 24.
2. Find all factorizations of 24.
Solution:
1. The factors of 24 are all of the divisors of 24.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
2. The factorizations of 24 are as follows:
1 ∙ 24, 2 ∙ 12, 3 ∙ 8, and 4 ∙ 6
Prime and Composite Numbers:
A natural number that has exactly two different factors, itself and 1, is called a prime number.
•
•
Examples:
The number 1 is not a prime number.
A natural number, other than 1, that is not prime is composite.
Determine whether the numbers listed below are prime, composite, or neither.
8
13
24
33
89
Solution:
18
18 is divisible by 2.
Therefore, it is
composite.
13
13 is divisible by only
one and itself.
Therefore, it is prime.
24
24 is divisible by 2.
Therefore it is
composite.
List of Primes from 2 to 100
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.
89
89 is divisible by only
one and itself.
Therefore, it is prime.
Prime Factorization:
Each composite number has a unique prime factorization.
The prime factorization of a number is the product of factors where each factor is a prime number.
Examples:
1.
2.
84
Find the prime factorization for each of the following numbers.
300
Solutions:
1.
84 is divisible by 2. So, we have 2 ∙ 42. The number 42 is also divisible by 2. Now,
we have 2 ∙ 2 ∙ 21. The number 21 is divisible by 3. This gives us 2 ∙ 2 ∙ 3 ∙ 7. Since
the factors above are all prime numbers, we have found the prime factorization for
84.
84 = 2 ∙ 2 ∙ 3 ∙ 7 or 22 ∙ 3 ∙ 7
2.
300 is divisible by 3. So, we have 3 ∙ 100. The number 100 is divisible by 2. Now,
we have 3 ∙ 2 ∙ 50. The number 50 is also divisible by 2. This gives us 3 ∙ 2 ∙ 2 ∙ 25.
The number 25 is divisible by 5. So, we have 3 ∙ 2 ∙ 2 ∙ 5 ∙ 5. Since the factors are all
prime numbers, we have found the prime factorization for 300.
300 = 3 ∙ 2 ∙ 2 ∙ 5 ∙ 5 or 22 ∙ 3 ∙ 52