For this strange trick you need to know what a MODULUS is.
If you divide 14 by 4 you get an answer of 3 plus a remainder of 2.
With some bits of maths, the remainder is
the important bit, and if you take a MODULUS, that means you're getting the remainder.
Here's how we'd write
down how to get the remainder of dividing 14 by 4: [14]mod4 =2
Here are a few other examples:
[10]mod3 =1 [23]mod7 =2 [23]mod3 =2 [100]mod9 =1
[27]mod26 =1 [26]mod27 =26
And finally [28]mod4 =0 because if you divide 28 by 4 there's no remainder!
What about [435]mod139?
Eeek! You don't really want to be dividing by 139, so what you can do is just keep
subtracting 139 until you can't go any further. At first you get 435139= 296 then 296139 = 157 then 157139=18. That's it! You can't go any further so [435]mod139 = 18.
The 105 Trick!

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 176105 =71.
Daniel Fretwell for telling us about this amazing trick! 