Pair of Positive Integers with Sum 91 and HCF 13

1. Understanding the Problem:
Given that the sum of two positive integers is 91 and their highest common factor (HCF) is 13, we need to find all possible pairs of positive integers that satisfy these conditions.

2. Relationship between Sum and HCF:
When two numbers have a HCF, it means that the largest number that divides both of them is their HCF. In this case, the HCF is 13. So, the two numbers must be multiples of 13.

3. Expressing the Numbers:
Let the two numbers be 13x and 13y, where x and y are positive integers. The sum of these two numbers is 91, so we have the equation:
13x + 13y = 91
x + y = 7

4. Finding the Pairs:
Since x and y are positive integers and their sum is 7, the possible pairs of x and y that satisfy this condition are:
(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)

5. Calculating the Numbers:
Substitute the values of x and y back into 13x and 13y to find the pairs of positive integers:
(13*1, 13*6) = (13, 78)
(13*2, 13*5) = (26, 65)
(13*3, 13*4) = (39, 52)
(13*4, 13*3) = (52, 39)
(13*5, 13*2) = (65, 26)
(13*6, 13*1) = (78, 13)

6. Conclusion:
Therefore, the pairs of positive integers whose sum is 91 and HCF is 13 are:
(13, 78), (26, 65), (39, 52), (52, 39), (65, 26), (78, 13)