(x, y) is a pair of positive integers such that HCF (x, y) + LCM (x, y) = 187 and x > y.

Which one of the following is correct in respect of the Question and the Statements given below?

Statement 1: There are three possible values of HCF (x, y).

Statement 2: The minimum value of (x + y) is 37.

Statement 3: There are 7 pairs of (x, y) that satisfy the given conditions.

Correct option is D

Given:
(x, y) is a pair of positive integers
HCF of x and y plus LCM of x and y is equal to 187
Also, x is greater than y
We are given three statements:
There are three possible values of HCF(x, y)
The minimum value of (x + y) is 37
There are 7 such (x, y) pairs that satisfy the condition
We are to evaluate which of the statements are correct.
Concept:
Let:
HCF of x and y = h
Then x = h × m, y = h × n
Where m and n are coprime integers, and since x > y, m > n
The LCM of x and y will be:
LCM = (x × y) / HCF = (h × m × h × n) / h = h × m × n
Now, the equation becomes:
HCF + LCM = h + h × m × n = h × (1 + m × n)
We are given:
h × (1 + m × n) = 187
This equation is key. We’ll now use it to find valid h values and count the valid (m, n) pairs for each.

Solution:
Step 1: Factor 187
187 = 11 × 17
Factors of 187: 1, 11, 17, 187

Try each factor of 187 as a possible value of h.

Step 2: Try h = 1
Then 1 + m × n = 187 → m × n = 186

Now, find all coprime pairs (m, n) such that:

m × n = 186

m > n

HCF(m, n) = 1

Factor pairs of 186:

(186, 1), (93, 2), (62, 3), (31, 6)

Check which of these are coprime:

(186, 1): coprime

(93, 2): coprime

(62, 3): coprime

(31, 6): coprime

Total valid pairs for h = 1: 4

Step 3: Try h = 11
Then 1 + m × n = 187 / 11 = 17
→ m × n = 16

Factor pairs of 16:

(16, 1), (8, 2), (4, 4)

Check for coprimality:

(16, 1): coprime

(8, 2): not coprime

(4, 4): not coprime

Total valid pairs for h = 11: 1

Step 4: Try h = 17
Then 1 + m × n = 187 / 17 = 11
→ m × n = 10

Factor pairs of 10:

(10, 1), (5, 2)

Both pairs are coprime.

Total valid pairs for h = 17: 2

Step 5: Try h = 187
Then 1 + m × n = 187 / 187 = 1
→ m × n = 0 — Invalid

Final Count:
h = 1 → 4 valid pairs

h = 11 → 1 valid pair

h = 17 → 2 valid pairs
Total = 4 + 1 + 2 = 7 valid pairs

Check Each Statement:

Statement 1: There are three possible values of HCF(x, y)
Correct — we found valid values of h as 1, 11, and 17
So Statement 1 is correct

Statement 2: The minimum value of x + y is 37
Try the pair (31, 6), from h = 1 and (m, n) = (31, 6)

x = 1 × 31 = 31

y = 1 × 6 = 6

x + y = 37

This is the minimum among all valid pairs, so
Statement 2 is correct

Statement 3: There are 7 valid pairs (x, y)
We found exactly 7 valid coprime (m, n) pairs, hence 7 valid (x, y) pairs
So Statement 3 is correct

Final Answer: D — All three statements are correct