SOLUTION: Two years ago my age was a multiple of 6. Last year it was a multiple of 5. I am less then 50 years old. How old am I now?

You are 26.  2 years ago you were 24, which is a multiple of 6
Last year you were 25, which is a multiple of 5. 

To work it out, let your age be N:

N - 2 = 6p
N - 1 = 5q
    N < 50

Solve the two equations for N

N = 6p + 2
N = 5q + 1

Equate the two expressions for N

6p + 2 = 5q + 1

     1 = 5q - 6p

Write 6 in terms of 5 as 5+1

     1 = 5q - (5+1)p
     1 = 5q - 5p - p
 
Divide thru by 5

   1/5 = q - p - p/5

Isolate the fractions:

p/5 - 1/5 = q - p

The right side is a positive integer, therefore
so is the left side. Let that positive integer be A

p/5 - 1/5 = A       also q - p = A
    p - 1 = 5A 
        p = 5A + 1

 Since q - p = A

q - (5A + 1) = A
  q - 5A + 1 = A
       q + 1 = 6A
           q = 6A - 1

So we have

p = 5A + 1
q = 6A - 1

Since N < 50 and 

N = 6p + 2

6p + 2 < 50
    6p < 48
     p < 8

and since p = 5A + 1

5A + 1 < 8
    5A < 7
     A < 7/5

Since A is a positive integer, 
it can only be 1.  So A = 1

 So p = 5A - 1 = 5(1) - 1 = 5 - 1 = 4 


So  

N = 6p + 2
N = 6(4)  + 2
N = 24 + 2
N = 26
 
Edwin