Question

Help please

problems number 3 and 4​

Answer 1

If we check prime numbers, we can see that only 2 and 5 produce a fraction with a finite number of decimal digits:

`(1)/(2) = 0.5,\quad (1)/(5) = 0.2`

All other primes produce fractions with periodic, infinite decimal expansions.

So, if the prime factorization of the denominator only contains 2 and 5, the fraction will have a finite decimal expansion.

In fact:

`(4)/(7) = 0.\overline{571428}`

because the prime factorization of the denominator contains 7.

`(5)/(6) = 0.8\overline{3}`

because the prime factorization of the denominator contains 2, but also 3.

`(25)/(12) = 2.08\overline{3}`

because the prime factorization of the denominator contains 2, but also 3.

`(19)/(20) = 0.95`

because the prime factorization of the denominator only contains 2 and 5.