+0

0
333
2

Identify which repeat and which terminate by not calculating. How do you know?

1. 1/40

2. 1/24

3. 2/125

4. 2/99

5. 7/300

6. 7/200

Oct 26, 2017

#1
+95859
+2

In a reduced form.....If we can express the denominator solely as a power of 2 or a power of 5 - or both - the fraction will terminate....else..it repeats.....

1/ 40  terminates  = 1 / [ 2^3 * 5]

1/ 24 does not =  1 / [ 2^3 * 3 ]

2/125 terminates  = 2 / 5^3

2/ 99 does not  =  2 / [ 3^2 * 11]

7/ 300  does not =  7 / [ 2^2 * 5^2 * 3 ]

7 / 200  terminates  =  7 / [ 2^3 * 5^2]

Here's the provision for the reduced form

3 / 300     appears to not terminate....however

3 / 300  =   1 / 100  =    1 /  [  2^2 * 5^2 ]    so this actually does terminate  !!!

And that's all there is to it  !!!!

Oct 26, 2017
#2
+96956
+1

Thanks Chris,

To put it another way..

If the decimal is to terminate then

The prime factors of the denominator must ONLY  be the prime factors of 10

(which are  2 and 5 )

----------------

I assume this can be extended to other bases.

So

1/5 base 8 is a recurring decimal because the prime factors of 5 are not the prime factors 8

That works.

I just did the division

and got

$$\frac{1}{5_8}=0.14\bar6\bar3$$   (base8)