Download Activity #34

Math 104 - Cooley
Math For Elementary Teachers I
OCC
Activity #34 – The Divisors of a Natural Number
California State Content Standard – Number Sense – Grade Four
4.0 Students know how to factor small whole numbers:
4.1 Understand that many whole numbers break down in different ways (e.g., 12 = 4 x 3 = 2 x 6 = 2 x 2 x 3).
Theorem – The Divisors of a Natural Number
Let n  p1a1  p 2a2    p rar be the prime-power representation of n. Then m divides n if, and only if,
m  p1b1  p 2b2    p rbr , where 0  b1  a1 , 0  b2  a 2 , …, 0  br  a r . Moreover, the number of factors, or
divisors, of n is given by N  (a1  1)(a 2  1)    (a r  1) .
 Example:
List all the divisors of 360.
Solution: First off, let’s find the prime-power representation of 360: 360  2 3  3 2  51
Second, let’s find the total number of factors:
Total number of factors, N  (3  1)  (2  1)  (1  1)  4  3  2  24
So, there are 24 different factors.
Lastly, according to the theorem, the divisors or factors must be all the numbers of the form 2 r  3 s  5 t
with 0  r  3 , 0  s  2 , and 0  t  1 . Now, make a systematic list of the divisors as follows:
1  2 0  30  5 0
2  21  3 0  5 0
4  2 2  30  5 0
8  2 3  30  5 0
3  2 0  31  5 0
6  21  31  5 0
12  2 2  31  5 0
24  2 3  31  5 0
9  2 0  32  50
18  21  3 2  5 0
36  2 2  3 2  5 0
72  2 3  3 2  5 0
5  2 0  3 0  51
10  21  30  51
20  2 2  3 0  51
40  2 3  3 0  51
15  2 0  31  51
30  21  31  51
60  2 2  31  51
120  2 3  31  51
45  2 0  3 2  51
90  21  3 2  51
180  2 2  3 2  51
360  2 3  3 2  51
So, the total 24 different factors are:
{ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 }
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Math 104 - Cooley
Math For Elementary Teachers I
OCC
Activity #34 – The Divisors of a Natural Number
 Exercises:
1)
Find the total number of factors for the number 2 3  7 5  4  3 2 .
2)
Find the total number of factors for the number 3 2  114  5 2  33
3)
List all the factors of 144.
4)
List al the factors of 1000.
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