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Joined: 10 Jul 2016
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n is an integer from 1 to 50, what is the probability that n(n+1) is n  [#permalink]

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Difficulty:   85% (hard)

Question Stats: 51% (01:55) correct 49% (02:34) wrong based on 147 sessions

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n is an integer from 1 to 50, what is the probability that n(n+1) is not divisible by 4?

A. 12/25
B. 13/25
C. 14/25
D. 15/25
E. 16/25
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Math Expert V
Joined: 02 Aug 2009
Posts: 7755
Re: n is an integer from 1 to 50, what is the probability that n(n+1) is n  [#permalink]

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siddharthsinha123 wrote:
n is an integer from 1 to 50, what is the probability that n(n+1) is not divisible by 4?

A. 12/25
B. 13/25
C. 14/25
D. 15/25
E. 16/25

For n(n+1) to be multiple of 4, either n or n+1 should be multiple of 4, as ONLY one of n or n+1 will be even and other will be ODD..

Till 50 , multiple of 4 are 50/4=12.25 so 12 multiples are there..
These 12 can be both n and n+1, so total 2*12=24..

Since n can take values from 1 to 50..
So n(n+1) will take 50 values..

Therefore ways when none of n and n+1 will be multiple of 4 is 50-24=26..
Prob of picking these =26/50=13/25

B
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Math Expert V
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Posts: 57193
Re: n is an integer from 1 to 50, what is the probability that n(n+1) is n  [#permalink]

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Senior SC Moderator V
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n is an integer from 1 to 50, what is the probability that n(n+1) is n  [#permalink]

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siddharthsinha123 wrote:
n is an integer from 1 to 50, what is the probability that n(n+1) is not divisible by 4?

A. 12/25
B. 13/25
C. 14/25
D. 15/25
E. 16/25

Multiples of x in the range = [ (Last multiple of x in the range - First multiple of x in the range ) / x ]+ 1

$$n = 4, 8, 12, 16, ..., 48$$

Multiples of x in the range = [ (48 - 4) / 4 ]+ 1 = 12

$$(n+1)$$ to be a multiples of 4, n must be 3, 7, 11, 15, ...., 47

$$n(n+1)$$ to be a multiple of 3, n can take 12 + 12 = 24 value

P = (favourable outcome) / (total number of outcomes)

P = 24 / 50

Probability that n(n+1) is not divisible by 4 = 1 - (24/50) = 13/25
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Re: n is an integer from 1 to 50, what is the probability that n(n+1) is n  [#permalink]

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siddharthsinha123 wrote:
n is an integer from 1 to 50, what is the probability that n(n+1) is not divisible by 4?

A. 12/25
B. 13/25
C. 14/25
D. 15/25
E. 16/25

We can use the following equation:

1 = P(n(n+1) is divisible by 4) + P(n(n+1) is NOT divisible by 4)

Thus:

P(n(n+1) is NOT divisible by 4) = 1 - P(n(n+1) is divisible by 4)

Let’s determine the probability that n(n+1) is divisible by 4. If n(n+1) is divisible by 4, then either n is divisible by 4 or n +1 is divisible by 4. Calculating the number of values of n divisible by 4 is the same as calculating the number of multiples of 4 between 1 and 50 inclusive. To calculate this, we can use this formula:

(largest multiple of 4 - smallest multiple of 4)/4 + 1

(48 - 4)/4 + 1

44/4 + 1 = 11 + 1 = 12

Thus, there are 12 multiples of 4 between 1 and 50 inclusive. That is, n can be any one of these 12 multiples of 4 so that n(n + 1) will be divisible by 4.

Similarly, if (n + 1) is a multiple of 4, n(n + 1) also will be divisible by 4. Since we know that there are 12 values of n that are multiples of 4, there must be another 12 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 = 47, n + 1 = 48, and thus n(n+1) is a multiple of 4.

We can see that there are 12 values of n that are multiples of 4, and 12 more values of n for (n + 1) to be a multiple of 4. Since there are 50 total integers from 1 to 50, inclusive, the probability of selecting a value of n so that n(n+1) is a multiple of 4 is:

24/50 = 12/25

Thus, the probability that n(n+1) IS NOT a multiple of 4 is 1 - 12/25 = 13/25.

Answer: B
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_________________ Re: n is an integer from 1 to 50, what is the probability that n(n+1) is n   [#permalink] 04 Aug 2019, 12:16
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