The number of pairs of positive integers whose sum is 99 and whose HCF is 9 is 5.

Quick reasoning

Let the two numbers be aaa and bbb with:

  • a+b=99a+b=99a+b=99
  • HCF(a,b)=9\text{HCF}(a,b)=9HCF(a,b)=9.

Since the HCF is 9, write:

  • a=9ma=9ma=9m
  • b=9nb=9nb=9n
    with HCF(m,n)=1\text{HCF}(m,n)=1HCF(m,n)=1.

Then:

9m+9n=99⇒m+n=11.[][]9m+9n=99\Rightarrow m+n=11.[][]9m+9n=99⇒m+n=11.[][]

All positive integer pairs (m,n)(m,n)(m,n) with m+n=11m+n=11m+n=11 are:

(1,10),(2,9),(3,8),(4,7),(5,6).[],(2,9),(3,8),(4,7),(5,6).[],(2,9),(3,8),(4,7),(5,6).[][]

Each of these pairs is coprime, so all are valid.

Corresponding (a,b)=(9m,9n)(a,b)=(9m,9n)(a,b)=(9m,9n) pairs are:

  • (9,90)(9,90)(9,90)
  • (18,81)(18,81)(18,81)
  • (27,72)(27,72)(27,72)
  • (36,63)(36,63)(36,63)
  • (45,54)(45,54)(45,54).

Thus, there are 5 such pairs.