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A = {x : x in N, GCD(x,36) =1, x < 36), ...

`A = {x : x in N, GCD(x,36) =1, x < 36),`
` B = {y : y in N, G.C.D.(y, 40)=1, y lt 40}` (G.C.D. stands greatest common divisors). Write in roaster form.

Text Solution

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The correct Answer is:
To solve the problem, we need to find the two sets A and B in roster form based on the given conditions. ### Step 1: Define Set A Set A is defined as: \[ A = \{ x \in \mathbb{N} : \text{GCD}(x, 36) = 1, x < 36 \} \] ### Step 2: Identify Natural Numbers Less Than 36 We will list all natural numbers less than 36: \[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35 \] ### Step 3: Find GCD with 36 Next, we need to find which of these numbers have a GCD of 1 with 36. The prime factorization of 36 is: \[ 36 = 2^2 \times 3^2 \] A number has a GCD of 1 with 36 if it does not share any prime factors with 36. Therefore, we will check each number: - **1**: GCD(1, 36) = 1 - **2**: GCD(2, 36) = 2 (not included) - **3**: GCD(3, 36) = 3 (not included) - **4**: GCD(4, 36) = 4 (not included) - **5**: GCD(5, 36) = 1 - **6**: GCD(6, 36) = 6 (not included) - **7**: GCD(7, 36) = 1 - **8**: GCD(8, 36) = 4 (not included) - **9**: GCD(9, 36) = 9 (not included) - **10**: GCD(10, 36) = 2 (not included) - **11**: GCD(11, 36) = 1 - **12**: GCD(12, 36) = 12 (not included) - **13**: GCD(13, 36) = 1 - **14**: GCD(14, 36) = 2 (not included) - **15**: GCD(15, 36) = 3 (not included) - **16**: GCD(16, 36) = 4 (not included) - **17**: GCD(17, 36) = 1 - **18**: GCD(18, 36) = 18 (not included) - **19**: GCD(19, 36) = 1 - **20**: GCD(20, 36) = 4 (not included) - **21**: GCD(21, 36) = 3 (not included) - **22**: GCD(22, 36) = 2 (not included) - **23**: GCD(23, 36) = 1 - **24**: GCD(24, 36) = 12 (not included) - **25**: GCD(25, 36) = 1 - **26**: GCD(26, 36) = 2 (not included) - **27**: GCD(27, 36) = 9 (not included) - **28**: GCD(28, 36) = 4 (not included) - **29**: GCD(29, 36) = 1 - **30**: GCD(30, 36) = 6 (not included) - **31**: GCD(31, 36) = 1 - **32**: GCD(32, 36) = 4 (not included) - **33**: GCD(33, 36) = 3 (not included) - **34**: GCD(34, 36) = 2 (not included) - **35**: GCD(35, 36) = 1 ### Step 4: List Elements of Set A The elements of set A are: \[ A = \{ 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35 \} \] ### Step 5: Define Set B Set B is defined as: \[ B = \{ y \in \mathbb{N} : \text{GCD}(y, 40) = 1, y < 40 \} \] ### Step 6: Identify Natural Numbers Less Than 40 We will list all natural numbers less than 40: \[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 \] ### Step 7: Find GCD with 40 The prime factorization of 40 is: \[ 40 = 2^3 \times 5 \] We will check each number for GCD with 40: - **1**: GCD(1, 40) = 1 - **2**: GCD(2, 40) = 2 (not included) - **3**: GCD(3, 40) = 1 - **4**: GCD(4, 40) = 4 (not included) - **5**: GCD(5, 40) = 5 (not included) - **6**: GCD(6, 40) = 2 (not included) - **7**: GCD(7, 40) = 1 - **8**: GCD(8, 40) = 8 (not included) - **9**: GCD(9, 40) = 1 - **10**: GCD(10, 40) = 10 (not included) - **11**: GCD(11, 40) = 1 - **12**: GCD(12, 40) = 4 (not included) - **13**: GCD(13, 40) = 1 - **14**: GCD(14, 40) = 2 (not included) - **15**: GCD(15, 40) = 5 (not included) - **16**: GCD(16, 40) = 8 (not included) - **17**: GCD(17, 40) = 1 - **18**: GCD(18, 40) = 2 (not included) - **19**: GCD(19, 40) = 1 - **20**: GCD(20, 40) = 20 (not included) - **21**: GCD(21, 40) = 1 - **22**: GCD(22, 40) = 2 (not included) - **23**: GCD(23, 40) = 1 - **24**: GCD(24, 40) = 8 (not included) - **25**: GCD(25, 40) = 5 (not included) - **26**: GCD(26, 40) = 2 (not included) - **27**: GCD(27, 40) = 1 - **28**: GCD(28, 40) = 4 (not included) - **29**: GCD(29, 40) = 1 - **30**: GCD(30, 40) = 10 (not included) - **31**: GCD(31, 40) = 1 - **32**: GCD(32, 40) = 8 (not included) - **33**: GCD(33, 40) = 1 - **34**: GCD(34, 40) = 2 (not included) - **35**: GCD(35, 40) = 5 (not included) - **36**: GCD(36, 40) = 4 (not included) - **37**: GCD(37, 40) = 1 - **38**: GCD(38, 40) = 2 (not included) - **39**: GCD(39, 40) = 1 ### Step 8: List Elements of Set B The elements of set B are: \[ B = \{ 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39 \} \] ### Final Answer Thus, the sets in roster form are: \[ A = \{ 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35 \} \] \[ B = \{ 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39 \} \]
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