Question 20

A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the from $\frac{\mathrm{p/}}{q}$ ? Give reason

Solution:

To express the decimal number 327.7081 in the form of $$\frac{\mathrm{p/}}{q}$$ and analyze the prime factors of $$q$$, we can follow these steps:

Step 1: Convert the Decimal to a Fraction
The decimal number is 327.7081. To convert this into a fraction, we can express it as:
$$327.7081=\frac{\mathrm{3277081/}}{10000}$$
Here, we multiply the number by 10,000 to eliminate the decimal point (since there are four digits after the decimal).

Step 2: Identify $$p$$ and $$q$$
From the fraction $$\frac{\mathrm{3277081/}}{10000}$$:
– $$p=3277081$$
– $$q=10000$$

Step 3: Factor $$q$$
Next, we need to find the prime factors of $$q$$:
$$10000={10}^4$$
Now, we can express 10 in terms of its prime factors:
$$10=2\times 5$$
Thus, we can rewrite $$10000$$ as:
$$10000=(2\times 5{)4}^{}={24}^{}\times {54}^{}$$

Step 4: Identify the Prime Factors of $$q$$
From the factorization of $$q$$:
$$q={24}^{}\times {54}^{}$$
The prime factors of $$q$$ are:
– 2
– 5

Conclusion
Therefore, when the decimal number 327.7081 is expressed in the form $$\frac{\mathrm{p/}}{q}$$, the prime factors of $$q$$ are 2 and 5.