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Let [tex]$A = \{n \in \mathbb{N} : \text{H.C.F.}(n, 45) = 1\}$[/tex] and let [tex]$B = \{2k : k \in \{1, 2, \ldots, 100\}\}$[/tex].

Then, the sum of all the elements of [tex][tex]$A \cap B$[/tex][/tex] is [tex]$\quad$[/tex] .



Answer :

GinnyAnswer
To find the sum of all elements in the intersection of sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we follow these steps:

1. Define Set [tex]\(A\)[/tex]:
- Set [tex]\(A\)[/tex] consists of natural numbers [tex]\(n\)[/tex] from 1 to 100 such that the highest common factor (H.C.F.) of [tex]\(n\)[/tex] and 45 is 1. This means that [tex]\(n\)[/tex] should not be divisible by any of the prime factors of 45. The prime factorization of 45 is [tex]\(45 = 3^2 \times 5\)[/tex]. Therefore, [tex]\(n\)[/tex] should not be divisible by 3 or 5.

The elements of [tex]\(A\)[/tex] are:
[tex]\[ \{1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 86, 88, 89, 91, 92, 94, 97, 98\} \][/tex]

2. Define Set [tex]\(B\)[/tex]:
- Set [tex]\(B\)[/tex] consists of even numbers from 2 to 200, i.e., numbers of the form [tex]\(2k\)[/tex] where [tex]\(k\)[/tex] is an integer from 1 to 100.

The elements of [tex]\(B\)[/tex] are:
[tex]\[ \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200\} \][/tex]

3. Find the Intersection [tex]\(A \cap B\)[/tex]:
- To find the intersection, we identify the common elements between [tex]\(A\)[/tex] and [tex]\(B\)[/tex].

The elements of [tex]\(A \cap B\)[/tex] are:
[tex]\[ \{2, 4, 8, 14, 16, 22, 26, 28, 32, 34, 38, 44, 46, 52, 56, 58, 62, 64, 68, 74, 76, 82, 86, 88, 92, 94, 98\} \][/tex]

4. Calculate the Sum of the Elements in [tex]\(A \cap B\)[/tex]:
- We sum all the elements in the intersection.

[tex]\[ 2 + 4 + 8 + 14 + 16 + 22 + 26 + 28 + 32 + 34 + 38 + 44 + 46 + 52 + 56 + 58 + 62 + 64 + 68 + 74 + 76 + 82 + 86 + 88 + 92 + 94 + 98 = 1364 \][/tex]

Thus, the sum of all the elements of [tex]\(A \cap B\)[/tex] is [tex]\(1364\)[/tex].

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