Let S = {1, 2, 3,…., 2022}. Then the probability, that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is:

<p>S = {1,2, 3, …., n, 2022}</p><p>HCF (n, 2022) = 1</p><p>2022 = 2 × 1011 -&gt;3 × 337</p><p>2022 = 2 × 3 × 337 (prime factorization)</p><p>Let n (A) = no members divisible by 2 = 1011</p><p>Let n (B) = no members divisible by 3 = 674</p><p>Let n (C) = no members divisible by 337 = 6</p><p><math><mrow><mi>n</mi><mrow><mo> (</mo><mrow><mi>A</mi><mo>∪</mo><mi>B</mi><mo>∪</mo><mi>C</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>n</mi><mrow><mo> (</mo><mrow><mi>A</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>n</mi><mrow><mo> (</mo><mrow><mi>B</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>n</mi><mrow><mo> (</mo><mrow><mi>C</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi>n</mi><mrow><mo> (</mo><mrow><mi>A</mi><mo>∩</mo><mi>B</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi>n</mi><mrow><mo> (</mo><mrow><mi>B</mi><mo>∩</mo><mi>C</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi>n</mi><mrow><mo> (</mo><mrow><mi>C</mi><mo>∩</mo><mi>A</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>n</mi><mrow><mo> (</mo><mrow><mi>A</mi><mo>∩</mo><mi>B</mi><mo>∩</mo><mi>C</mi></mrow><mo>)</mo></mrow></mrow></math></p><p>= 1011 + 674 + 6 – 337 – 2 – 3 + 1</p><p>= 1350</p><p><math><mrow><mi>n</mi><mrow><mo> (</mo><mrow><mi>A</mi><mo>∩</mo><mi>B</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mn>3</mn><mn>7</mn></mrow></math></p><p><math><mrow><mi>n</mi><mrow><mo> (</mo><mrow><mrow><mo> (</mo><mrow><mi>A</mi><mo>∪</mo><mi>B</mi><mo>∪</mo><mi>C</mi></mrow><mo>)</mo></mrow><mo>'</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mn>0</mn><mn>2</mn><mn>2</mn><mo>−</mo><mn>1</mn><mn>3</mn><mn>5</mn><mn>0</mn><mo>=</mo><mn>6</mn><mn>7</mn><mn>2</mn></mrow></math></p><p>Prob.&nbsp;<math><mrow><mo>=</mo><mfrac><mrow><mn>6</mn><mn>7</mn><mn>2</mn></mrow><mrow><mn>2</mn><mn>0</mn><mn>2</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mn>3</mn><mn>6</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>1</mn><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mn>1</mn><mn>2</mn></mrow><mrow><mn>3</mn><mn>3</mn><mn>7</mn></mrow></mfrac></mrow></math></p><p><img ></p>

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