What is the probability that the product of the numbers will be a multiple of 6?
Assuming 2 randomly selected positive integers are multiplied
together, I came up with a 5 in 12 chance (41 2/3 percent )
Here's how I figured it: take the first number and divide by six
and examine the remainder (called modulo 6). You have 6 different
possibilities (0,1,2,3,4,5), each with a 1 in six chance of
occurring.
If mod 6 = 0, then the first number is a multiple by 6, and it
doesn't matter what the 2nd number is; all products are multiples
of 6 as well.
If mod 6 = 1 or 5, then the 2nd number must be a multiple of six
to get the product to be multiple of 6. The chance of the 2nd
number a multiple of six is 1 in 6.
If mod 6 = 2, then these are numbers like 2,8,14,20... They are
even numbers, so if the second number is a multiple of 3, then the
product is multiple of 6. The chance of the 2nd number a multiple
of three is 1 in 3.
Mod 6 = 3, then first number is multiple of 3 (like 3,9,15,21),
and the second number must be even (chance is 1 in 2).
Mod 6 = 4, then first number is like 4,10,16,22 It is an even
number so the 2nd number must be multiple of 3 (chance is 1 in
3).
So in order how I described, we have (1 + 1/6 + 1/6 + 1/3 + 1/2
+ 1/3)*(1/6) which equals 5/12.