GreenlightTestPrep wrote:
If integer x is chosen at random from 50 to 149 inclusive, what is the probability that x³ - x is a multiple of 12?
Enter your answer as a
fraction The number of integers from x to y inclusive equals y - x + 1So, number of integers from 50 to 149 inclusive = 149 - 50 + 1 =
10012 = (2)(2)(3)
So, in order for (x - 1)(x)(x + 1) to be a multiple of 12, we need at least two 2's and one 3 "hiding" in the product (x - 1)(x)(x + 1)
Now notice that x³ - x = x(x² - 1) = x(x + 1)(x - 1) = (x - 1)(x)(x + 1)
Notice that x-1, x and x+1 are
three consecutive integersNice rule:
The product of k consecutive integers is divisible by k, k-1, k-2,...,2, and 1This means (x - 1)(x)(x + 1) is divisible by 3 for ALL integer values of x.
Since (x - 1)(x)(x + 1) is guaranteed to be divisible by 3, we just need to find
the probability that (x - 1)(x)(x + 1) is divisible by 4 Now notice that HALF of the products (x - 1)(x)(x + 1) are in the form (EVEN)(ODD)(EVEN) and half are in the form (ODD)(EVEN)(ODD)
Let's examine each case separately.
Case i: The product (x - 1)(x)(x + 1) is in the form (EVEN)(ODD)(EVEN)
We can write: (x - 1)(x)(x + 1) = (some multiple of
2)(ODD)(some multiple of
2) = (
2)(
2)(some integer) = (
4)(some integer)
This means (x - 1)(x)(x + 1) is a multiple of
4 every time x - 1 is EVEN
Since ALL products in the form (EVEN)(ODD)(EVEN) are divisible by 4 (and by 3), we can conclude that these products are divisible by 12.
Since even and odd numbers alternate, we can see that HALF of
100 possible products will be such that x - 1 is EVEN
In other words
50 products are such that x - 1 is EVEN (which means those products are divisible by 12 too)
Case ii: The product (x - 1)(x)(x + 1) is in the form (ODD)(EVEN)(ODD)
Also notice that the product can also have the form (ODD)(EVEN)(ODD)
SOME of the products in this form are divisible by 4, and SOME of the products are not divisible by 4.
Products in the form (ODD)(EVEN)(ODD) will be divisible by 4 when the middle number is divisible by 4.
For example, (59)(60)(61) is divisible by 4, since 60 is divisible by 4.
Likewise, (61)(62)(63) is NOT divisible by 4, since 62 is NOT divisible by 4.
And (63)(64)(65) is divisible by 4, since 64 is divisible by 4.
And (67)(68)(69) is NOT divisible by 4, since 68 is NOT divisible by 4.
As you can see, HALF of the products in the form (ODD)(EVEN)(ODD) are divisible by 4, and half of the products the form (ODD)(EVEN)(ODD) are NOT divisible by 4
50 of the 100 products are in the form (ODD)(EVEN)(ODD)
And HALF of those 50 are divisible by 4.
So,
25 products in the form (ODD)(EVEN)(ODD) are divisible by 4 (which means those products are divisible by 12 too)
So the total number of products that are divisible by 12 =
50 +
25 =
75The probability that (x - 1)(x)(x + 1) is divisible by 12 =
75/
100 = 3/4
Cheers,
Brent