If 3p7q8 is divisible by 44,list all the possible pairs of values p and q.
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)
