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Gmatblast, Kpadma is right in his reasoning because all you need to consider is the units digit of those numbers. No matter how large those numbers are, the proportion, through an infinite set of numbers, will be the same.
What I am not certain about is when Kpadma says: "Numbers that yield only 1,3,7 and 9 as unit digit
when two numbers are multiplied are number that have
1,3,7, and 9 in their unit digit."
The original question mentions about four numbers being multiplied together,...
Here is how I thought about this problem (considering that it is whole numbers and not digits)
Now the probability = Number of ways in which 4 numbers are selected from the set of numbers that meet our criteria / Number of ways to select 4 numbers from all the possible numbers
For numerator, we have infinite possibilities from which to select 4 numbers (inf C 4)
But I think the key is to understand that there are 4 numbers with 1,3,7,9 in first 10 numbers. Similarly there are 40 numbers with 1,3,7,9 as unit digits in first 100 numbers. and so on...
What I am not certain about is when Kpadma says: "Numbers that yield only 1,3,7 and 9 as unit digit when two numbers are multiplied are number that have 1,3,7, and 9 in their unit digit."
I think what kpadma meant was if you take any four numbers with unit digit = 1,3,7, 9, the product of all the 4 numbers will always have 1,3,7,9 as unit digit. So that is OK. If any of the 4 numbers have a unit digit other that 1,3,7 or 9, the product will also have unit digit other than 1,3,7 or 9