mitko20m wrote:
If an integer n is to be selected at random from 1 to 100, inclusive, what is probability n(n+1) will be divisible by 4?
A. 1/4
B. 1/3
C. 1/2
D. 2/3
E. 3/4
We are given that an integer n is to be selected at random from 1 to 100, inclusive, and we must determine the probability that n(n+1) will be divisible by 4.
Since probability = favorable outcomes/total outcomes, and we know that the total number of outcomes is 100, since there are 100 integers from 1 to 100, inclusive, we need to determine the number of values of n such that n(n+1) is divisible by 4.
First, we can determine the number of values of n that are divisible by 4, that is, the number of multiples of 4 that are between 1 and 100, inclusive. To calculate this, we can use the formula:
(Largest multiple of 4 – smallest multiple of 4)/4 + 1
(100 - 4)/4 + 1
96/4 + 1
24 + 1 = 25
Thus, there are 25 multiples of 4 between 1 and 100 inclusive. That is, the value of n can be any one of these 25 multiple of 4 so that n(n + 1) will be divisible by 4.
Similarly, if n + 1 is a multiple of 4, n(n + 1) will be also be divisible by 4. Since we know that there are 25 values of n that are multiples of 4, there must be another 25 values of n such that n + 1 is a multiple of 4. Let’s expand on this idea:
When n = 3, n + 1 = 4, and thus n(n+1) is a multiple of 4…..
When n = 23, n + 1 = 24, and thus n(n+1) is a multiple of 4…..
When n = 99, n + 1 = 100, and thus n(n+1) is a multiple of 4.
So we can see that there are 25 values of n that are multiples of 4 and 25 more values of n for n + 1 that are multiples of 4. Thus, the probability of selecting a value of n so that n(n+1) is a multiple of 4 is:
50/100 = 1/2
Answer: C