A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form , where is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by or ? Give reasons.

Understand the Question A terminating decimal that ends at the decimal place can be written with digits after the decimal point, where the digit is non-zero. Multiplying by shifts the decimal point places to the right, producing an integer whose last digit is non-zero (hence not divisible by ). Writing the fraction in lowest terms involves cancelling common factors of and from the denominator . Since is not divisible by , it cannot be divisible by both and at the same time. Step 1 · Express the number in the form Let the rational number be . Since its decimal expansion terminates with the last non-zero digit in the decimal place, it can be represented as where , digits , and . Multiply by Let , which is an integer. Since the last digit of is , is not divisible by . Therefore Step 2 · Analyze the denominator in lowest form We have To write in lowest terms, we divide the numerator and denominator by their greatest common divisor. Since is not divisible by , cannot have both and as prime factors simultaneously. There are three cases: Case 1: is divisible by (or powers of ) but not by . Then no power of is cancelled from the denominator. Thus, the denominator in lowest form retains the factor and is divisible by . Case 2: is divisible by (or powers of ) but not by . Then no power of is cancelled from the denominator. Thus, the denominator in lowest form retains the factor and is divisible by . Case 3: is divisible by neither nor . Then neither nor is cancelled, so the denominator in lowest form remains , which is divisible by both and . In all cases, the denominator in lowest form is divisible by or . Answer > Yes, it is necessary that the denominator in lowest form is divisible by or . Common Mistakes Assuming both factors can cancel: Overlooking that prevents from sharing both factor and factor with at the same time. Confusing 'Divisible by 10' with 'Divisible by 2 or 5': A number not divisible by can still be divisible by alone (e.g., ) or alone (e.g., ), but never both.

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Question 10

A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form $\dfrac{p}{10^4}$ , where $p$ is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by $2^4$ or $5^4$ ? Give reasons.

Solution

Understand the Question

A terminating decimal that ends at the $4^{\text{th}}$ decimal place can be written with $4$ digits after the decimal point, where the $4^{\text{th}}$ digit is non-zero.
Multiplying by $10^4$ shifts the decimal point $4$ places to the right, producing an integer whose last digit is non-zero (hence not divisible by $10$ ).
Writing the fraction in lowest terms involves cancelling common factors of $2$ and $5$ from the denominator $10^4 = 2^4 \times 5^4$ . Since $p$ is not divisible by $10$ , it cannot be divisible by both $2$ and $5$ at the same time.

Step 1 · Express the number in the form $\dfrac{p}{10^4}$

Let the rational number be $x$ .

Since its decimal expansion terminates with the last non-zero digit in the $4^{\text{th}}$ decimal place, it can be represented as $x = a.d_1d_2d_3d_4$ where $a \in \mathbb{Z}$ , digits $d_1, d_2, d_3, d_4 \in \{0, 1, 2, \dots, 9\}$ , and $d_4 \neq 0$ .

Multiply $x$ by $10^4$ $10^4 x = a d_1 d_2 d_3 d_4$

Let $p = 10^4 x$ , which is an integer.

Since the last digit of $p$ is $d_4 \neq 0$ , $p$ is not divisible by $10$ .

Therefore $x = \dfrac{p}{10^4}$

Step 2 · Analyze the denominator in lowest form

We have $x = \dfrac{p}{10^4} = \dfrac{p}{2^4 \times 5^4}$

To write $x$ in lowest terms, we divide the numerator and denominator by their greatest common divisor.

Since $p$ is not divisible by $10$ , $p$ cannot have both $2$ and $5$ as prime factors simultaneously. There are three cases:

Case 1: $p$ is divisible by $2$ (or powers of $2$ ) but not by $5$ . Then no power of $5$ is cancelled from the denominator. Thus, the denominator in lowest form retains the factor $5^4$ and is divisible by $5^4$ .

Case 2: $p$ is divisible by $5$ (or powers of $5$ ) but not by $2$ . Then no power of $2$ is cancelled from the denominator. Thus, the denominator in lowest form retains the factor $2^4$ and is divisible by $2^4$ .

Case 3: $p$ is divisible by neither $2$ nor $5$ . Then neither $2^4$ nor $5^4$ is cancelled, so the denominator in lowest form remains $2^4 \times 5^4$ , which is divisible by both $2^4$ and $5^4$ .

In all cases, the denominator in lowest form is divisible by $2^4$ or $5^4$ .

Answer

Yes, it is necessary that the denominator in lowest form is divisible by $2^4$ or $5^4$ .

Common Mistakes

Assuming both factors can cancel: Overlooking that $10 \nmid p$ prevents $p$ from sharing both factor $2$ and factor $5$ with $10^4$ at the same time.
Confusing 'Divisible by 10' with 'Divisible by 2 or 5': A number not divisible by $10$ can still be divisible by $2$ alone (e.g., $12$ ) or $5$ alone (e.g., $15$ ), but never both.

Question 10

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