Question

Seven times b plus 8 gives 88​

Answer 1

Answer:

The summands that can be used is 8, 88, 888. (8888 onwards are clearly not eligible)

If the max summand is 888 all answers have shown that 888+88+8+8+8 is a way. We cant have 888 and two 88 as it will exceed 1000, we can consider the sum as 888+k*8 where k is a natural number and show that k=14. So 888+ 14 copies of 8 is also another way.

If the max is 8 a sum of 125 copies of 8 is also a way.

Lastly consider the case where max summand is 88. Consider the diophantine equation where 1000=88k+8m where k, m are integers. Then we have 125=11k+m. By Chinese Remainder theorem(?) or any other facts, this eqn is solvable, (one solution is (11,4)) and any other solutions is in the form (11-n,4+11n) for all n integers. Lastly we need k>0 and m positive (our premise is we have 88 as a summand so k cannot be 0) (m has to be positive as the qn does not allow subtraction) that gives 10 distinc solutions. So 10 ways here.