(I'm Overwhelmed) Arithmetic Progress Quiz #7 - Question 9

I didn’t write this one, but solving this one on-the-fly,

\frac{a}{b} = \frac{29}{8}

so possible values of a and b would be multiples of 29 and 8 respectively:

  • a = 29, b = 8 → remainder = 5
  • a = 58, b = 16 → remainder = 10
  • a = 87, b = 24 → remainder = 15
  • a = 116, b = 32 → remainder = 20

so the remainder has to be a multiple of 5 → options D and F are correct.

Another option would be to show that, for k being a positive integer,

a = 29 b + 5

and hence multiplying by k on both sides

ka = k(29b + 5)
ka = 29kb + 5k

→ remainder when ka is divided by kb is 5k.