Help please problems number 3 and 4

Question

asked Oct 20, 2020 199k views

1 Answer

Aseem Sharma · Answer 1 · 2020-10-26T20:04:54+0000

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.