The sum of two numbers is 528 and their H.C.F. is 33. How many distinct pairs of positive integers satisfy these conditions?

Introduction / Context:
This question checks whether you can use the concept of H.C.F. to express numbers in a common form and then count the number of valid pairs. It combines number theory with basic counting and factorization and is a common type in competitive exams where you must find how many pairs of numbers fit certain conditions.

Given Data / Assumptions:

• The sum of two positive integers is 528.
• Their H.C.F. is 33.
• We must find the number of pairs of such numbers.

Concept / Approach:
If two numbers have H.C.F. 33, we can write them as 33a and 33b, where a and b are co-prime positive integers. Their sum is 33a + 33b = 33(a + b). Since the sum is 528, we can find a + b. Every pair of co-prime positive integers (a, b) with that sum corresponds to a valid pair of numbers. So our task reduces to counting co-prime pairs whose sum is a fixed number.

Step-by-Step Solution:
Step 1: Let the two numbers be 33a and 33b with H.C.F.(a, b) = 1.Step 2: Sum of the numbers is 33a + 33b = 33(a + b).Step 3: Given sum = 528, so 33(a + b) = 528.Step 4: Divide to get a + b = 528 / 33 = 16.Step 5: We now need pairs of positive integers (a, b) such that a + b = 16 and gcd(a, b) = 1.Step 6: Possible positive integer pairs (a, b) with sum 16 are (1, 15), (2, 14), (3, 13), (4, 12), (5, 11), (6, 10), (7, 9), and (8, 8).Step 7: Check gcd for each: gcd(1, 15) = 1, gcd(2, 14) = 2, gcd(3, 13) = 1, gcd(4, 12) = 4, gcd(5, 11) = 1, gcd(6, 10) = 2, gcd(7, 9) = 1, gcd(8, 8) = 8.Step 8: Valid co-prime pairs are (1, 15), (3, 13), (5, 11), and (7, 9).Step 9: Therefore there are 4 valid pairs of (a, b), giving 4 pairs of numbers (33a, 33b).

Verification / Alternative check:
Example pair: a = 1, b = 15 gives numbers 33 and 495 with H.C.F. 33 and sum 528.Similarly for (3, 13), (5, 11), and (7, 9) you get other valid pairs with H.C.F. 33 and sum 528. No other co-prime pairs exist, so there are exactly 4 possible pairs.

Why Other Options Are Wrong:
Options b (6), c (8), and d (12) overcount because they treat pairs with common factors as valid or fail to enforce the co-prime condition required by the H.C.F. representation. Only the four pairs listed respect both the co-prime condition and the sum requirement.

Common Pitfalls:
Students often count all pairs with the correct sum without checking if a and b are co-prime. Others may mistakenly think that (a, b) and (b, a) are separate pairs, but the question is about unordered pairs of numbers. A disciplined approach of listing and checking gcd values prevents these mistakes.

Final Answer:
The number of such pairs is 4.