CBSE Guess > Papers > Important Questions > Class X > 2010 > Maths > Mathematics By Mr. Kulvinder CBSE CLASS X Q.1. Based on Euclid’s algorithm: a = bq + r Using Euclid’s algorithm: Find the HCF of 825 and 175. Explanation: Step 1. Since 825>175. Divide 825 by 175. We get, quotient = 4 and remainder = 125. This can be written as 825 = 175 x 4 + 125 Solution: This is how a student should write answer in his answer sheet: Since 825>175, we apply division lemma to 825 and 175 to get Problems for practice;
Q.2. Based on Showing that every positive integer is either of the given forms: Prove that every odd positive integer is either of the form 4q + 1 or 4q + 3 for some integer q. Euclid’s division lemma a= bq + r. Comparing this with the given integers (4q + 1) we get that b should be 4.If we divide any number by 4 possible remainders are 0, 1, 2 or 3 because fourth number will again be divided by 4. Ex 12 ÷ 4, r=0; 13÷4, r=1; 14÷ 4, r=2; 15÷4, r=3; 16÷4 once again r= 0.Hence possible remainders are 0, 1, 2 or 3. If r = 0, then we get a = 4q, If r = 1 we get a= 4q + 1 and so on till r = 3 which will give a= 4q + 3. Since we want only odd integers our choices are 4q + 1 and 4q + 3. Solution: Let a be any odd positive integer (first line of problem) and let b = 4. Using division Lemma we can write a = bq + r, for some integer q, where 0≤r<4. So a can be 4q, 4q + 1, 4q + 2 or 4q + 3. But since a is odd, a cannot be 4q or 4q + 2. Therefore any odd integer is of the form 4q + 1 or 4q + 3. Practice questions:
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