Fully worked, step-by-step solutions to A-Level Proof by contradiction questions. See exactly how to solve problems on primes, number-theory, rationals, contradiction.
Interpret the task
We test the integers above 7 for primality.
Identify the first prime
This is the first integer above 7 with no proper divisors.
State the answer
This is the required prime.
Interpret the task
We test the integers above 13 for primality.
Identify the first prime
This is the first integer above 13 with no proper divisors.
State the answer
This is the required prime.
Interpret the task
We test the integers above 20 for primality.
Identify the first prime
This is the first integer above 20 with no proper divisors.
State the answer
This is the required prime.
Take the given positive rational
Assume, for contradiction, it were the smallest positive rational.
Compare with the original
The half is smaller, so the original was not the smallest.
State a smaller positive rational
Its existence contradicts minimality.
Take the given positive rational
Assume, for contradiction, it were the smallest positive rational.
Compare with the original
The half is smaller, so the original was not the smallest.
State a smaller positive rational
Its existence contradicts minimality.
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