Two random integers, a and b, are independently chosen, with replacement, from 1 to 1000, inclusive. What is the probability that both 2^a 2^b and 3^a 3^b are multiples of 5

The probability is zero

Step-by-step explanation:

For a number to be a multiple of 5, the number must end with either 0 or 5

These are the only numbers in which we can have 5 as a factor of

Now when we raise 2 to any number, there are only some sets of possible results

These numbers are only multiples of 2

These can be;

2, 4, 6 and 8

No root of two has its end in 0 or 5, so none has 5 has its factor

Let us now look at 3

The multiples of 3 when raised are;

3,9,1,7,

As we can see, at no point do we have 5 or 0 as the result

So since we have none of both as ending 5 or 0, the probability we are looking for is zero