2. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
3. 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? Sol. Hints: Find the HCF of 616 and 32
4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. [Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]
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.
6. Consider the numbers 4n, where n is a natural number. Check whether there is any value of nfor which 4n ends with the digit zero.
7. Find the LCM and HCF of 6 and 20 by the prime factorization method.
8. Find the HCF of 96 and 404 by the prime factorization method. Hence, find their LCM.
9. Find the HCF and LCM of 6, 72 and 120, using the prime factorization method.
10. Find the value of y if the HCF of 210 and 55 is expressible in the form 210 x 5 + 55y
11. Prove that no number of the type 4K + 2 can be a perfect square.