SOLUTION: What least number, when divided by 26,36 and 56, leaves the remainders 6,16,36 respectively ?

n = 26a+6 = 36b+16 = 56c+36

We solve the Diophantine equation

36b+16 = 56c+36
  9b+4 = 14c+9
    9b = 14c+5

9 is the least integer in that equation
in absolute value, so write all the other
integers in terms of a near multiple
of 9.  So we write 14 as 18-4, and 5 is
near 9 so we just leave it as it is.

9b = (18-4)c+5
9b = 18c-4c+5
Divide through by 9
b = 2c-4c/9+5/9

Get the fraction on the left and other terms
on the right:

4c/9-5/9 = 2c-b

The right side is an integer, say P, so

4c/9-5/9 = 2c-b = P

4c/9-5/9 = P

4c-5 = 9P

4 is the least integer in that equation
in absolute value, so write all the other
integers in terms of their nearest multiple
of 4.  So we write 5 as 4+1, write 9 as 8+1.

4c-(4+1) = (8+1)P

4c-4-1 = 8P+P
 
Divide through by 4

c-1-1/4 = 2P+P/4

Get the fraction on the right, other terms
on the left:

c-1-2P = P/4+1/4

The left side is an integer, say Q, so

c+1-2P = P/4+1/4 = Q

P/4+1/4 = Q
    P+1 = 4Q
      P = 4Q-1
c-1-2P = Q
c-1-2(4Q-1) = Q
c-1-8Q+2 = Q
  c+1-8Q = Q
       c = 9Q-1

9b = 14c+5
9b = 14c+5
9b = 14(9Q-1)+5
9b = 126Q-14+5
9b = 126Q-9
 b = 14Q-1

26a+6 = 36b+16
26a = 36b+10
13a = 18b+5
13a = 18(14Q-1)+5
13a = 252Q-18+5
13a = 252Q-13

13 is the least integer in that equation
in absolute value, so write all the other
integers in terms of their nearest multiple
of 13.  So we write 252 as 247+5.

13a = (247+5)Q-13
13a = 247Q+5Q-13
Divide through by 13
  a = 19Q+5Q/13-1

Get the fraction on the right and other
terms left:

a+1-19Q = 5Q/13

The left side is an integer, say R, so

a+1-19Q = 5Q/13 = R

5Q/13 = R

5Q = 13R

We could continue as before but this equation is
simpler and we see that the smallest integers Q 
and R that solve that are Q=13 and R=5.

So a+1-19Q = R
   a+1-19(13) = 5
   a+1-247 = 5
    a-246 = 5
        a = 251

And since n = 26a+6
          n = 26(251)+6
          n = 6526+6
          n = 6532

Checking:

    251          181          116
26)6532      36)6532      56)6532
   52           36           56
   133          293           93
   130          288           56 
     32           52          372
     26           36          336
      6           16           36  <-remainders check!

Edwin