Let's analyze the problem step-by-step:
Problem:
Find the number of pairs of positive integers (a,b)(a,b)(a,b) such that:
- a+b=99a+b=99a+b=99
- HCF(a,b)=9\text{HCF}(a,b)=9HCF(a,b)=9
Step 1: Express aaa and bbb in terms of their HCF
Since the highest common factor (HCF) of aaa and bbb is 9, we can write:
a=9x,b=9ya=9x,\quad b=9ya=9x,b=9y
where xxx and yyy are positive integers such that HCF(x,y)=1\text{HCF}(x,y)=1HCF(x,y)=1 (because all common factors are already factored out as 9).
Step 2: Use the sum condition
Given:
a+b=99a+b=99a+b=99
Substitute a=9xa=9xa=9x and b=9yb=9yb=9y:
9x+9y=99 ⟹ x+y=999=119x+9y=99\implies x+y=\frac{99}{9}=119x+9y=99⟹x+y=999=11
So, x+y=11x+y=11x+y=11 and HCF(x,y)=1\text{HCF}(x,y)=1HCF(x,y)=1.
Step 3: Find pairs (x,y)(x,y)(x,y) such that x+y=11x+y=11x+y=11 and
gcd(x,y)=1\gcd(x,y)=1gcd(x,y)=1
Since xxx and yyy are positive integers, possible pairs (x,y)(x,y)(x,y) with sum 11 are:
(1,10),(2,9),(3,8),(4,7),(5,6),(6,5),(7,4),(8,3),(9,2),(10,1)(1,10),(2,9),(3,8),(4,7),(5,6),(6,5),(7,4),(8,3),(9,2),(10,1)(1,10),(2,9),(3,8),(4,7),(5,6),(6,5),(7,4),(8,3),(9,2),(10,1)
Now, check which pairs have gcd(x,y)=1\gcd(x,y)=1gcd(x,y)=1:
- gcd(1,10)=1\gcd(1,10)=1gcd(1,10)=1 ✔
- gcd(2,9)=1\gcd(2,9)=1gcd(2,9)=1 ✔
- gcd(3,8)=1\gcd(3,8)=1gcd(3,8)=1 ✔
- gcd(4,7)=1\gcd(4,7)=1gcd(4,7)=1 ✔
- gcd(5,6)=1\gcd(5,6)=1gcd(5,6)=1 ✔
- gcd(6,5)=1\gcd(6,5)=1gcd(6,5)=1 ✔
- gcd(7,4)=1\gcd(7,4)=1gcd(7,4)=1 ✔
- gcd(8,3)=1\gcd(8,3)=1gcd(8,3)=1 ✔
- gcd(9,2)=1\gcd(9,2)=1gcd(9,2)=1 ✔
- gcd(10,1)=1\gcd(10,1)=1gcd(10,1)=1 ✔
All pairs have gcd=1\gcd =1gcd=1.
Step 4: Count the number of valid pairs
There are 10 such pairs.
Step 5: Consider if pairs (a,b)(a,b)(a,b) and (b,a)(b,a)(b,a) are distinct
If the order matters (i.e., (a,b)(a,b)(a,b) and (b,a)(b,a)(b,a) are different), the answer is 10. If the order does not matter (pairs are unordered), then pairs like (1,10)(1,10)(1,10) and (10,1)(10,1)(10,1) are the same, so the number of distinct pairs is:
102=5\frac{10}{2}=5210=5
Final Answer:
- If ordered pairs are counted: 10 pairs
- If unordered pairs are counted: 5 pairs
Summary:
Condition| Number of pairs
---|---
Ordered pairs (a,b)(a,b)(a,b)| 10
Unordered pairs {a,b}\{a,b\}{a,b}| 5
If the problem does not specify, usually unordered pairs are considered, so the number of pairs is 5.
