A number is divisible by 44 if it is divisible by both 4 and 11.

44 = 4x11

We will use the rules for divisibility by 4 and 11 to restrict the values of 'p' and 'q'.

For divisibility by 4, the last two digits should be divisible by 4

For divisibility by 11, the difference between the sum of alternate digits should be divisible by 11

Divisibility by 4:

08-- q = 0

28-- q = 2

48-- q = 4

68-- q = 6

88-- q = 8

Possible values for q are 0,2,4,6,8

Divisibility by 4:

Alternate sum: (3+7+8) - (p-q) = 18 - (p+q), should be 0,11, or 22

Possible values for p+q are 7 or 18

Now, list pairs for each q value where p is a single digit:

For each q value (from Step 1), p = 7−q and p = 18−q (if p is a single digit)

q = 0

p=7 + p=18 (not possible) p = 7 or p = 18 (only p = 7 is allowed) So, (p, q) = (7, 0)

q = 2

p=5 or p=16 (only p=5) (p, q) = (5, 2)

q = 4

p = 3 or p = 14 (only p = 3) (p, q) = (3, 4)

q = 6

p = 1 or p = 12 (only p = 1) (p, q) = (1, 6)

q = 8

p = −1 or p = 10 (not possible, p must be positive and single digit).

So, no solution.

So, the possible pairs of values p and q are

(7,0)
(5,2)
(3,4)
(1,6)