The decimal expansion of $\frac{p}{q}$ will be terminating precisely when the prime factors of $q$ are only 2, only 5, or both 2 and 5. Can you explain why? [Page No. 58]

Because our number system is base 10, and 10 = $2 \times 5$.

A decimal terminates exactly when the fraction can be rewritten with a denominator that is a power of 10 (10, 100, 1000, ...). Every power of 10 is $2n \times 5n$ made of only 2s and 5s. So if the denominator (in lowest terms) is built only from 2s and/or 5s, you can multiply the
top and bottom by enough extra 2s or 5s to turn the denominator into a power of 10, and the decimal stops. If the denominator has any other prime factor (like 3 or 7), no such multiplication can ever produce a pure power of 10, so the division never ends it repeats instead.

Example: $\frac{3}{20}$, with $20 = 2^2 \times 5$. Multiply top and bottom by 5: $\frac{15}{100} = 0.15$, terminates.

Contrast $\frac{1}{3}$: the denominator 3 can never become a power of 10, so $\frac{1}{3} = 0.333...$ repeats.