Show that a fraction r=a / b in lowest terms has a finite...

Show that a fraction $r=a / b$ in lowest terms has a finite decimal expansion if and only if $$b=2^{n} 5^{m} \quad$ for some $n, m \geq 0$$ Hint: Observe that $r$ has a finite decimal expansion when $10^{N} r$ is an integer for some $N \geq 0$ (and hence $b$ divides $10^{N} )$ .

Key Concepts

Finite Decimal Expansion

A finite decimal expansion of a rational number means that the decimal representation terminates after a finite number of digits. This occurs when the number can be expressed as an integer when multiplied by a certain power of the base (in base 10, a power of 10), meaning the fraction fully converts to a decimal without an infinite repeating cycle.

Prime Factorization

Analyzing the prime factorization of the denominator of a fraction is crucial because it determines whether the denominator can be 'canceled out' by a power of the base. In any numeral system, only those prime factors that also appear in the base can result in a terminating expansion of the fraction.

Lowest Terms

Expressing a fraction in lowest terms ensures that the numerator and denominator share no common factors. This is important because it isolates the intrinsic prime factors of the denominator, which decides if the fraction will have a finite decimal expansion based solely on the structure of the denominator.

Divisibility by Powers of the Base

A key concept is that a fraction will have a terminating decimal expansion if its denominator divides some power of the base. In the decimal system, this means the denominator must be a divisor of a number like 10^N for some integer N, which is possible only when the denominator’s prime factors are limited to 2 and 5, the prime factors of 10.

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