A rational number has 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 $\frac{p}{10^4}$, 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.

If the last non-zero digit is at the 4th place, the decimal has the form $0 \cdot d_1d_2d_3d_4$ (with $d_4 \neq 0$).
 Multiplying by $10^4$ gives the integer

$$p = d_1d_2d_3d_4.$$

So, the number equals $\frac{p}{10^4}$.

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

When written in lowest form, the denominator need NOT be divisible by $2^4$ or $5^4$.

Counter-example: $0.0625 = \frac{625}{10000}$.

Simplify: HCF(625, 10000) = 625

$$\rightarrow 0.0625 = \frac{1}{16} = \frac{1}{2^4}.$$

Denominator = $2^4$ (divisible by $2^4$ but not $5^4$).

Another: $0.0016 = \frac{16}{10000} = \frac{1}{625} = \frac{1}{5^4}$

(divisible by $5^4$ but not $2^4$).

Another: $0.0003 = \frac{3}{10000}$.

HCF(3, 10000) = 1 → denominator = 10000 = $2^4 \times 5^4$ (divisible by both).

So, the denominator must be of the form $2^a \times 5^b$ where $0 \leq a, b \leq 4$ and $\max(a, b) = 4$. It is not necessary that both $2^4$ and $5^4$ divide it.