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Q5  Use Euclid’s division lemma to show that the cube of any positive integer is of the form
       9m, 9m + 1 or 9m + 8.

Let x be any positive integer. It can be written in the form 3q + r where  and r = 0, 1 or 2 Case 1: For r = 0 we have x3 = (3q)3 x3 = 27q3 x3 = 9(3q3) x3 = 9m Case 2: For r = 1 we have x3 = (3q+1)3 x3 = 27q3 + 27q2 + 9q + 1 x3 = 9(3q3 + 3q2 +q) + 1 x3 = 3m + 1 Case 3: For r = 2 we have x3 = (3q+2)3 x3 = 27q3 + 54q2 + 36q + 8 x3 = 9(3q3 + 6q2 +4q) + 8 x3 = 3m + 8 Hence proved.

Q4  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.] 

       

Let x be any positive integer. It can be written in the form 3q + r where  and r = 0, 1 or 2 Case 1: For r = 0 we have x2 = (3q)2 x2 = 9q2 x2 = 3(3q2) x2 = 3m Case 2: For r = 1 we have x2 = (3q+1)2 x2 = 9q2 + 6q +1 x2 = 3(3q2 + 2q) + 1 x2 = 3m + 1 Case 3: For r = 2 we have x2 = (3q+2)2 x2 = 9q2 + 12q +4 x2 = 3(3q2 + 4q + 1) + 1 x2 = 3m + 1 Hence proved.

Q3  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?

The maximum number of columns in which they can march = HCF (32, 616) Since 616 > 32, applying Euclid's Division Algorithm we have Since remainder  0 we again apply Euclid's Division Algorithm Since 32 > 8  Since remainder  = 0 we conclude, 8 is the HCF of  616 and  32.  The maximum number of columns in which they can march is 8.

Q2   Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is
        some integer.

Let p be any positive integer. It can be expressed as p = 6q + r where  and but for r = 0, 2 or 4 p will be an even number therefore all odd positive integers can be written in the form 6q + 1, 6q + 3 or 6q + 5.

Q1 (3)   Use Euclid’s division algorithm to find the HCF of :  867 and 255

867 > 225. Applying Euclid's Division algorithm we get since remainder  0 we apply the algorithm again. since 255 > 102 since remainder  0 we apply the algorithm again. since 102 > 51 since remainder = 0 we conclude the HCF of  867 and 255 is 51.

Q1 (2)   Use Euclid’s division algorithm to find the HCF of :  196 and 38220

38220 > 196. Applying Euclid's Division algorithm we get since remainder = 0 we conclude the HCF of  38220 and 196 is 196.

Q1 (1)   Use Euclid’s division algorithm to find the HCF of :  135 and 225

225 > 135. Applying Euclid's Division algorithm we get since remainder  0 we again apply the algorithm since remainder  0 we again apply the algorithm since remainder = 0 we conclude the HCF of 135 and 225 is 45.
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