SOLUTION: we have an unknown number of coin.If you make 77 strings of them, you are 50 coins short, but if you make 78 strings , it is exact.how many coins are there? [hint:use diophantine

Let n be the answer.

So n is a multiple of 78, and therefore 

n = 78p, for some natural number q

So if 50 is added to n, the result is a multiple of 77

Therefore 

   n + 50 = 77q for some integer

Substitute 78p for n

(1)   78p + 50 = 77q

The smaller coefficient of a letter in absolute value is 77

So we rewrite 78 and 50 in terms of their nearest multiples of 77
We rewrite 78 as 77+1 and 50 as 77-27

  (77+1)p + 77-27 = 77q

77p + p + 77 - 27 = 77q

Divide through by 77

  p +  + 1 -  = q

Isolate the fractional terms

- = q - p - 1

The right side is an integer. Let that integer be A.

(2)     q - p - 1 = A

and so is the left side

            - = A  

Clear of fractions 

            p - 27 = 77A

                 p = 77A + 27

Substitute in (2)

         q - p - 1 = A
q - (77A + 27) - 1 = A
  q - 77A - 27 - 1 = A
      q - 77A - 28 = A
                 q = 28 + 77A

So the solution to (1) is (p,q) = (77A+27,28+77A)

n = 78p = 78(77A+27) = 6006A+2106

The smallest solution is when A = 0, 

n = 2106.

But there are infinitely many solutions, as A can be chosen as 
any positive integer.

Edwin