**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

Guest Oct 26, 2017

#1**+2 **

In a* reduced form.*....If we can express the denominator

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 !!!!

CPhill
Oct 26, 2017

#2**+1 **

Thanks Chris,

I had not thought about that before.

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)

How about that!

Thanks Chris

Melody
Oct 26, 2017