Chapter 4.2, Problem 51E

To determine

To prove property of co-primes that if a, b and b, c are co-primes, then a, c may not be co-primes.

Answer to Problem 51E

No

Explanation of Solution

Given Information:a, b and b, c are co-primes, then a, c may not be co-primes.

Formula Used:if a, b and b, c are co-primes, then a, c may not be co-primes. This is because a number which is not divisible with the other number, could be divisible with a new number.

Calculation:

To prove this lets take an example. Let’s assume a=16, b=35 and c=64.

As it is given a, b and b, c are co-prime numbers.

For a and b:

The factors of 16 are: 1, 2, 4, 8, 16

The factors of 35 are: 1, 5, 7, 35

Then the greatest common factor is 1.

For b and c:

The factors of 35 are: 1, 5, 7, 35

The factors of 64 are: 1, 2, 4, 8, 16, 32, 64

Then the greatest common factor is 1.

Now for a and c:

The factors of 16 are: 1, 2, 4, 8, 16

The factors of 64 are: 1, 2, 4, 8, 16, 32, 64

Then the greatest common factor is 16.

Now, since the GCF of a, c i.e. 16, 64 IS NOT 1, therefore, a and c are not co-prime numbers.

Hence Proved.