Use Euclid's division algorithm to find the $HCF$ of $135$ and $225$.

Use Euclid's division algorithm to find the $HCF$ of $867$ and $255$.

Without actually performing the long division,state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion: $\frac{129}{2^{2} \cdot 5^{7} \cdot 7^{5}}$

Use Euclid's division lemma to show that the cube of any positive integer is of the form $9m, 9m+1$ or $9m+8$.

Show that any positive odd integer is of the form $4q+1$ or $4q+3,$ where $q$ is some integer.

The following real number has a decimal expansion as given below. Decide whether it is rational or not. If it is rational,and of the form $\frac{p}{q},$ what can you say about the prime factors of $q$?
$43.123456789$