Question

without actually performing the long division state whether 13/3125 and 13/343 will have a terminating decimal​

Answer 1

Step-by-step explanation:

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Answer 2

SOLUTION

TO CHECK

To identify the below two fractions have terminating decimal expansion or a non - terminating repeating decimal decimal expansion

$\displaystyle \sf{ (i) \: \: \: \frac{13}{3125} \: \: }$

$\displaystyle \sf{ (ii) \: \: \: \frac{13}{343} \: \: }$

CONCEPT TO BE IMPLEMENTED

$\displaystyle \sf{\: Fraction \: = \: \frac{Numerator}{Denominator} \: }$

A fraction is said to have terminating decimal expansion if the 2 & 5 are the only prime factors of the denominator

Otherwise the fraction is said to have non terminating decimal expansion

EVALUATION

CHECKING FOR OPTION (i)

$\displaystyle \sf{ (i) \: \: \: \frac{13}{3125} \: \: }$

Here denominator = 3125

Now

$\sf{ 3125 = 5 \times 5 \times 5 \times 5 \times 5\: \: }$

Since 5 is the only prime number present in prime factorisation of 3125

Hence this fraction has terminating decimal expansion

CHECKING FOR OPTION (ii)

$\displaystyle \sf{ (ii) \: \: \: \frac{13}{343} \: \: }$

Here denominator = 343

Now $\sf{ 343 = 7 \times 7 \times 7\: }$

Since 7 is the prime number present in prime factorisation of 343

Hence this fraction has non terminating decimal expansion

RESULT

$\boxed{ \displaystyle \sf{ (i) \: \frac{13}{3125} \: \: has \: terminating \: decimal \: expansion \: }}$

$\boxed{ \displaystyle \sf{ (ii) \: \frac{13}{343} \: \: has \:non \: terminating \: decimal \: expansion \: }}$

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