Question

In: Advanced Math

Use the method of exhaustion to prove the following statement: “For every prime number p between...

Use the method of exhaustion to prove the following statement: “For every prime number p between 30 and 58, 10 does not divide p − 9.”
Prove that 0.17461461 . . . is rational (digits 461 in the fractional part are periodically repeated forever).

Solutions

Expert Solution

First question:

Primes between and are

These are primes because and has no divisor other than (neither of divides ) implies is a prime, and has no divisor other than (neither of divides ) implies is a prime, and has no divisor other than (neither of divides ) implies is a prime, and has no divisor other than (neither of divides ) implies is a prime, and has no divisor other than (neither of divides 47) implies 47 is a prime, and has no divisor other than (neither of divides ) implies is a prime.

Now, for , we have

and divides none of . This proves the desired statement.

Second question:

We have

The series in the parenthesis is a geometric series with common ratio

Therefore,

which is a rational number.

Colby Messinger answered 1 year ago

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