Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 11I / c

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- Solution c

Solution

By lifting a primitive root to odd prime powers, it is enough to test primitivity modulo

25

. The positive integers below eight either fail to be units or have orders

ord_{25} (1) = 1, ord_{25} (4) = 10, ord_{25} (6) = 5, ord_{25} (7) = 4,

whereas

ord_{25} (2) = ord_{25} (3) = ord_{25} (8) = 20.

Therefore the three smallest required integers are

2, 3, 8 .

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