# Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:(i) $\frac{13}{3125}$.(ii) $\frac{17}{8}$.(iii) $\frac{64}{455}$.(iv) $\frac{15}{1600}$.(v) $\frac{29}{343}$.(vi) $\frac{23}{2^3\times5^2}$.(vii) $\frac{129}{2^2\times5^7\times7^{17}}$.(viii) $\frac{6}{15}$.(ix) $\frac{35}{50}$.(x) $\frac{77}{210}$.

#### Complete Python Prime Pack for 2023

9 Courses     2 eBooks

#### Artificial Intelligence & Machine Learning Prime Pack

6 Courses     1 eBooks

#### Java Prime Pack 2023

8 Courses     2 eBooks

To do:

Here, we have to check without actually performing the long division, whether the given rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

Solution:

If we have a rational number $\frac{p}{q}$, where $p$ and $q$ are co-primes and the prime factorization of $q$ is of the form $2^n.5^m$, where $n$ and $m$ are non-negative integers, then $\frac{p}{q}$ has a terminating expansion.

Now,

(i) In $\frac{13}{3125}$,

• $13$ and $3125$ are co-primes.
• $3125 = 5^5 \times 2^0$, which is in the form $2^n\ \times\ 5^m$.

So, $\frac{13}{3125}$ has a terminating decimal expansion.

(ii) In $\frac{17}{8}$,

• $17$ and $8$ are co-primes.
• $8= 2^3 \times 5^0$, which is in the form $2^n\ \times\ 5^m$.

So, $\frac{17}{8}$ has a terminating decimal expansion.

(iii) $\frac{64}{455}=\frac{64}{5\times7\times13}$

In $\frac{64}{455}$,

• $64$ and $455$ are co-primes.
• $455= 5 \times 7\times13$, which is not in the form $2^n\ \times\ 5^m$.

So, $\frac{64}{455}$ has a non-terminating repeating decimal expansion.

(iv) $\frac{15}{1600}=\frac{3\times5}{2^6\times5^2}$

$=\frac{3}{2^6\times5}$

In $\frac{3}{2^6\times5}$,

• $3$ and $2^6\times5$ are co-primes.
• $2^6\times5= 2^6 \times 5^1$, which is in the form $2^n\ \times\ 5^m$.

So, $\frac{15}{1600}$ has a terminating decimal expansion.

(v) $\frac{29}{343}=\frac{29}{7\times7\times7}$

In $\frac{29}{343}$,

• $29$ and $343$ are co-primes.
• $343= 7^3$, which is not in the form $2^n\ \times\ 5^m$.

So, $\frac{29}{343}$ has a non-terminating repeating decimal expansion.

(vi) In $\frac{23}{2^3\times5^2}$,

• $23$ and $2^3\times5^2$ are co-primes.
• $2^3 \times 5^2$ is in the form $2^n\ \times\ 5^m$.

So, $\frac{23}{2^3\times5^2}$ has a terminating decimal expansion.

(vii) In $\frac{129}{2^2\times5^7\times7^{17}}$:

• $129$ and $2^2\times5^7\times7^{17}$ are co-primes.
• $2^2 \times 5^7 \times 7^{17}$ is not in the form $2^n\ \times\ 5^m$.

So, $\frac{129}{2^2\times5^7\times7^{17}}$ has a non-terminating repeating decimal expansion.

(viii) $\frac{6}{15}=\frac{2\times3}{3\times5}$

$=\frac{2}{5}$

In $\frac{2}{5}$,

• $2$ and $5$ are co-primes.
• $5= 2^0 \times 5^1$, which is in the form $2^n\ \times\ 5^m$.

So, $\frac{6}{15}$ has a terminating decimal expansion.

(ix) $\frac{35}{50}=\frac{5\times7}{5\times10}=\frac{7}{10}$

In $\frac{7}{10}$:

• $7$ and $10$ are co-primes.
• $10 = 2^1 \times 5^1$, which is in the form $2^n\ \times\ 5^m$.

So, $\frac{35}{50}$ has a terminating decimal expansion.

(x) $\frac{77}{210}=\frac{7\times11}{7\times30}=\frac{11}{30}$

In $\frac{11}{30}$:

• $11$ and $30$ are co-primes.
• $30 = 2^1 \times 3^1 \times 5^1$, which is not in the form $2^n\ \times\ 5^m$.

So, $\frac{77}{210}$ has a non-terminating repeating decimal expansion.

Updated on 10-Oct-2022 13:19:30