Let a and b be two numbers such that a and b - a are co-primes and b and b + a are co- primes, respectively. Which of the followings is true?

Correct option is D

Given:

a and b-a are co-primes

b and b + a are co-primes

Formula Used:

Co-primes are the numbers which have highest common factor as 1.

Solution:

Given that HCF(a, b-a) =1

And HCF(b, b+a ) =1

Lets assume that a and b are not co-primes. This means there exists a common divisor d>1 such that a = dx and b = dy for some integer x and y

Substituting into the given conditions:

HCF(a, b-a) =1 = HCF(dx, dy-dx) = HCF(dx, d(y-x) = d $$\times$$​ HCF(x, y-x)

HCF(x, y-x) =$$\frac{1}{d}$$​​

Which is impossible since x and y are integers and d>1.

It leads to contradiction of a and b sharing divisor d>1

Hence, our assumption must be wrong and a and b are co-primes