THE MODULUS 105 trick

The 105 Trick!
Ask your friend to think of a mystery number Q between 1 and 105.
Then ask him to work out [Q]mod3 and tell you the answer. (In other words he has to divide his mystery number by 3 and tell you what the remainder is). Call the answer x
Ask him to work out [Q]mod5. Call the answer y
Ask him to work out [Q]mod7. Call the answer z
You then put the answers for x,y and z in this formula: [70x+21y+15z]mod105
...and you can work out Q !
Example: if the mystery number is 71 then...
71/3 = 23 remainder 2 so [71]mod3=2. This gives x=2.
71/5 = 14 remainder 1 so [71]mod5=1. This gives y=1.
71/7 = 10 remainder 1 so [71]mod7=1. This gives z=1.
Your friend has told you these values for x,y and z (but not the number 71 obviously).
Put these answers in [70x+21y+15z]mod105 to get
[70x2 + 21x1 + 15x1]mod105 = [140 + 21 + 15]mod105 = [176]mod105

Finally to work out [176]mod105 you just do 176-105 =71.