[spoiler='Proof
']First, Q,S,U,W and Y must be even numbers (they are the second digit of numbers that must be divisible by a number with a factor of 2.
Therefore the others must be odd.
- Y must = 0 as any number divisible by 10.
- T must = 5 as any number divisible by 5 must end in 0 or 5 (0 is taken by Y)
- U must = 4, for working down, the only number divisible by 6 that starts with 5 is 54
- V must = 9 for the only two numbers divisible by 7 that begin with 4 are 42 and 49, but V must be odd (see top)
- W must = 6 as the only number beginning with 9 that is divisible by 8 is