Q. Show that any positive odd integer is of the form $6q+1$, or $6q+3$, or $6q+5$, where $q$ is some integer.

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

Q.

Find the HCF of the following numbers by using Euclid's division algorithm.

$196$ and $38220$

Q. Use Euclids division lemma to show that the square of any positive integer is either of the form $3m$ or $3m+1$ for some integer $m$.

Q. An army contingent of $616$ members is to march behind an army band of $32$ members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Q. Use Euclids division algorithm to find the HCF of: $867$ and $225$

Q. Use Euclids division algorithm to find the HCF of:

Q. Check whether $6^n$ can end with the digit $0$ for any natural number $n$.

Q. There is a circular path around a sports field. Sonia takes $18$ minutes to drive one round of the field, while Ravi takes $12$ minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Q. Is Duttada’s general outlook (i) rational? (ii) moral? (iii) traditional? Choose the right word. Say why you think it right

Q. Given that HCF $(306$ and $657)=9$, find LCM of $(306$ and $657)$

Q. Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$HCF = product of the two numbers.
(i) $336$ and $54$

Q. Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i) $12,15$ and $21$ (ii) $17,23$ and $29$ (iii) $8,9$ and $25$

Q. Express each number as a product of its prime factors:

Q. Prove that the following are irrational:
(i) $\frac{1}{\sqrt{2}}$ (ii) $7\sqrt{5}$ (iii) $6+\sqrt{2}$

Q. From the third paragraph pick out
(i) words associated with cries of birds,
(ii) words associated with noise,
(iii) words suggestive of confusion and fear.

Q. Prove that $3+2\surd 5$ is irrational.

Q. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational and of the form $p$, you say about the prime factors of $q$?
(i) $43.123456789$ (ii) $\mathrm{0.120120012000120000.}..$ (iii) $43.\overline{123456789}$

Q. Write down the decimal expansions of the rational numbers which have terminating decimal expansions.
(i) $\frac{13}{3125}$ (ii) $\frac{17}{8}$ (iii) $\frac{15}{1600}$ (iv) $\frac{23}{2^3{5}^2}$ (v) $\frac{6}{15}$ (vi) $\frac{35}{50}$

Q. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i) $\frac{13}{3125}$ (ii) $\frac{17}{8}$ (iii) $\frac{64}{455}$ (iv) $\frac{15}{1600}$ (v) $\frac{29}{343}$