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Problem:
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Is the number 621 prime or composite?
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| Method:
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In the last lesson, we learned to find all factors
of a whole number to determine if it is prime or composite. We used the procedure listed below.
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To determine if a number is prime or composite, follow these steps:
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- Find all factors of the number.
- If the number has only two factors, 1 and itself, then it is prime.
- If the number has more than two factors, then it is composite.
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| The above procedure works very well for small numbers. However, it would be time-consuming to
find all factors of 621. Thus we need a better method for determining if
a large number is prime or composite.
Every number has one and itself as a factor. Thus, if we could find one factor of 621, other than
1 and itself, we could prove that 621 is composite. One way to find factors of large numbers
quickly is to use tests for divisibility.
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| Definition |
Example |
| One whole number is divisible by another if, after dividing, the
remainder
is zero.
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18 is divisible by 9 since 18 ÷ 9 = 2 with a remainder of 0.
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| If one whole number is divisible by another number, then the second number is a
factor of the first number.
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Since 18 is divisible by 9, 9 is a factor of 18.
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| A divisibility test is a rule for determining whether one whole number is
divisible by another. It is a quick way to find factors of large numbers.
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Divisibility Test for 3: if the sum of the digits of a number is divisible by 3, then the
number is divisible by 3.
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We can test for divisibility by 3 (see table above) to quickly find a factor of 621 other than 1
and itself. The sum of the digits of 621 is 6+2+1 = 9. This divisibility test and the definitions
above tell us that...
- 621 is divisible by 3 since the sum of its digits (9) is divisible by 3.
- Since 621 is divisible by 3, 3 is a factor of 621.
- Since the factors of 621 include 1, 3 and 621, we have proven that 621 has more than two factors.
- Since 621 has more than 2 factors, we have proven that it is composite.
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Let's look at some other tests for divisibility and examples of each.
| Divisibility Tests |
Example |
| A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8.
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168 is divisible by 2 since the last digit is 8.
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| A number is divisible by 3 if the sum of the digits is divisible by 3.
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168 is divisible by 3 since the sum of the digits is 15 (1+6+8=15), and 15 is divisible by 3.
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| A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
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316 is divisible by 4 since 16 is divisible by 4.
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| A number is divisible by 5 if the last digit is either 0 or 5.
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195 is divisible by 5 since the last digit is 5.
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| A number is divisible by 6 if it is divisible by 2
AND it is divisible by 3.
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168 is divisible by 6 since it is divisible by 2 AND it is divisible by 3.
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| A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
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7,120 is divisible by 8 since 120 is divisible by 8.
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| A number is divisible by 9 if the sum of the digits is divisible by 9.
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549 is divisible by 9 since the sum of the digits is 18 (5+4+9=18), and 18 is divisible by 9.
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| A number is divisible by 10 if the last digit is 0.
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1,470 is divisible by 10 since the last digit is 0.
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Let's look at some examples in which we test the divisibility of a single whole number.
| Example 1:
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Determine whether 150 is divisible by 2, 3, 4, 5, 6, 9
and 10.
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150 is divisible by 2 since the last digit is 0.
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150 is divisible by 3 since the sum of the digits is 6 (1+5+0 = 6), and 6 is divisible by 3.
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150 is not divisible by 4 since 50 is not divisible by 4.
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150 is divisible by 5 since the last digit is 0.
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150 is divisible by 6 since it is divisible by 2 AND by 3.
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150 is not divisible by 9 since the sum of the digits is 6, and 6 is not divisible by 9.
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150 is divisible by 10 since the last digit is 0.
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| Solution:
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150 is divisible by 2, 3, 5, 6, and 10.
![[IMAGE]](images/checkmark.gif)
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| Example 2:
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Determine whether 225 is divisible by 2, 3, 4, 5, 6, 9
and 10.
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225 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.
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225 is divisible by 3 since the sum of the digits is 9, and 9 is divisible by 3.
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225 is not divisible by 4 since 25 is not divisible by 4.
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225 is divisible by 5 since the last digit is 5.
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225 is not divisible by 6 since it is not divisible by both 2 and 3.
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225 is divisible by 9 since the sum of the digits is 9, and 9 is divisible by 9.
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225 is not divisible by 10 since the last digit is not 0.
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| Solution:
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225 is divisible by 3, 5 and 9.
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| Example 3:
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Determine whether 7,168 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
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7,168 is divisible by 2 since the last digit is 8.
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7,168 is not divisible by 3 since the sum of the digits is 22, and 22 is not divisible by 3.
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7,168 is divisible by 4 since 168 is divisible by 4.
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7,168 is not divisible by 5 since the last digit is not 0 or 5.
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7,168 is not divisible by 6 since it is not divisible by both 2 and 3.
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7,168 is divisible by 8 since the last 3 digits are 168, and 168 is divisible by 8.
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7,168 is not divisible by 9 since the sum of the digits is 22, and 22 is not divisible by 9.
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7,168 is not divisible by 10 since the last digit is not 0 or 5.
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| Solution:
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7,168 is divisible by 2, 4 and 8.
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| Example 4:
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Determine whether 9,042 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
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9,042 is divisible by 2 since the last digit is 2.
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9,042 is divisible by 3 since the sum of the digits is 15, and 15 is divisible by 3.
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9,042 is not divisible by 4 since 42 is not divisible by 4.
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9,042 is not divisible by 5 since the last digit is not 0 or 5.
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9,042 is divisible by 6 since it is divisible by both 2 and 3.
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9,042 is not divisible by 8 since the last 3 digits are 042, and 42 is not divisible by 8.
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9,042 is not divisible by 9 since the sum of the digits is 15, and 15 is not divisible by 9.
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9,042 is not divisible by 10 since the last digit is not 0 or 5.
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| Solution:
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9,042 is divisible by 2, 3 and 6.
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| Example 5:
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Determine whether 35,120 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
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35,120 is divisible by 2 since the last digit is 0.
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35,120 is not divisible by 3 since the sum of the digits is 11, and 11 is not divisible by 3.
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35,120 is divisible by 4 since 20 is divisible by 4.
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35,120 is divisible by 5 since the last digit is 0.
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35,120 is not divisible by 6 since it is not divisible by both 2 and 3.
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35,120 is divisible by 8 since the last 3 digits are 120, and 120 is divisible by 8.
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35,120 is not divisible by 9 since the sum of the digits is 11, and 11 is not divisible by 9.
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35,120 is divisible by 10 since the last digit is 0.
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| Solution:
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35,120 is divisible by 2, 4, 5, 8 and 10.
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| Example 6:
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Is the number 91 prime or composite? Use divisibility when possible to
find your answer.
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91 is not divisible by 2 since the last digit is not 0, 2, 4, 6 or 8.
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91 is not divisible by 3 since the sum of the digits (9+1=10) is not divisible by 3.
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91 is not evenly divisible by 4 (remainder is 3).
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91 is not divisible by 5 since the last digit is not 0 or 5.
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91 is not divisible by 6 since it is not divisible by both 2 and 3.
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91 divided by 7 is 13.
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| Solution:
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The number 91 is divisible by 1, 7, 13 and 91. Therefore 91 is composite since it has more
than two factors.
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