EXERCISE 1.1 I. Use Euclid’s division algorithm to find the HCF of: 135 and 225 h) 196 and 38230 ) 867 and 255 2. Show that any positive and integer is of the form 6q+1, or 6+), of by+5. where ou some integer 3. An army contingent of 616 members in 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? 4. Use Euclid’s division femrumu to show that the square of any positive integer is either of the form 3mor – for some integer m. Hint : Lexbe any positive integer then it is of the form 3, 34+ or 34+2. Now square cach of these and show that they can be rewritten in the form 3.0 or +1.) 5. Lise Euclid’ division lemma to show that the cube of any positive integer is of the form 9. Sim + 1 or Sm+8 1.3 The Fundamental Theorem of Arithmetic In your earlier classes, you have seen that any natural number can he written as a product of its prime factors. For instance, 2 = 2.4 = 2 2.253= 11 x 23, and so on. Now, let us try and look at natural numbers from the other direction. That is, can any natural number be obtained by multiplying prime numbers! Let us see. Take any collection of prime numbers, say 2. 3. 7. 11 and 23. If we multiply some or all of these numbers, allowing them to repeat as many times as we wish, we can produce a large collection of positive integers (In fact, infinitely many). Let us list a few 7×11* 23 = 1771 3x7x11 x 23 = 5313 2×3 x 7 x 11 x 23 = 10626 2′ x 3 x 7′ = 8232 22*3*7* 11 * 23=21252