Show that for each nautral number n, the fraction is in its lowestform.

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Solution

Using Euclidean Algorithm, we have

21n + 4 = 1(14n + 3) + (7n + 1)

14n + 3 = 2(7n + 1) + 1

7n + 1 = 1(7n + 1) + 0

∴ GCD (7n + 1, 1) = 1

Also, GCD (21n + 4, 14n + 3) = GCD (14n + 3, 7n + 1) = GCD (7n + 1, 1) = 1

Since GCD (21n + 4, 14n + 3) =1, therefore, the fraction is in its lowest form for each natural number n.

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