Prove that if \(p\) is a prime number and \(0

Short Answer

Expert verified

The largest number that can divide both a prime number \(p\) and any positive integer \(h\) less than \(p\) is 1. Hence, the greatest common divisor (gcd) of \(p\) and \(h\) is always 1.

Step by step solution

01

Understand the properties of prime numbers

Prime numbers are numbers that have only two factors: 1 and themselves. Therefore, any number less than a prime number, like \(h\) in this case, cannot be a factor of the prime number \(p\). This is because \(h\) is strictly less than \(p\), and by definition, \(p\) has no factors other than 1 and itself.

02

Consider the possible gcd values

The Greatest Common Divisor (gcd) of two numbers is the largest number that divides both of them without leaving a remainder. Since from step 1 we know \(h\) cannot be a factor of \(p\), the gcd of \(p\) and \(h\) cannot be \(h\) itself or any value greater than 1. The gcd also cannot be 0, as zero doesn't divide anything.

03

Conclude the proof

By the principle of elimination, the only possible gcd left is 1. Therefore, we can establish that gcd\((p, h)=1\).

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