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If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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If integer C is randomly selected from 20 to 99, inclusive. What is the probability that C^3 - C is divisible by 12 ?

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 1/3

Originally posted by shashankp27 on 05 Oct 2011, 09:44.
Last edited by Bunuel on 07 Jul 2013, 06:27, edited 2 times in total.
Edited the question.
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Re: divisible by 12 Probability  [#permalink]

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If integer C is randomly selected from 20 to 99, inclusive. What is the probability that c^3-c is divisible by 12?
A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 1/3

Two things:
1. There are total of 80 integers from 20 to 99, inclusive: 20, 21, ..., 99.
2. C^3-C=(C-1)*C*(C+1): we have the product of 3 consecutive integers, which is always divisible by 3. So the question basically is whether (C-1)*C*(C+1) is divisible by 4.

Next, the only way the product NOT to be divisible by 4 is C to be even but not a multiple of 4, in this case we would have (C-1)*C*(C+1)=odd*(even not multiple of 4)*odd.

Now, out of first the 4 integers {20, 21, 22, 23} there is only 1 even not multiple of 4: 22. All following groups of 4 will also have only 1 even not multiple of 4 (for example in {24, 25, 26, 27} it's 26, and in {96, 97, 98, 99} it's 98, always 3rd in the group) and as our 80 integers are entirely built with such groups of 4 then the overall probability that C is not divisible by 4 is 1/4. Hence the probability that it is divisible by 4 is 1-1/4=3/4.

nkhosh wrote:
Would you please explain (1) The formulas you used for finding the # of ODD and "Divisible by 4" outcomes?
AND: (2) Why are we looking for the number of ODD integers within 20-99? I'm confused b/c 23 for example is not divisible by 12....

Thank you for the great post 1. There are even # of consecutive integers in our range - 80 (from 20 to 99, inclusive). Out of even number of consecutive integers half is always odd and half is always even, thus numbers of odd integers in the given range is 40.

2. $$# \ of \ multiples \ of \ x \ in \ the \ range =$$
$$=\frac{Last \ multiple \ of \ x \ in \ the \ range \ - \ First \ multiple \ of \ x \ in \ the \ range}{x}+1$$.

So, for our case as 96 is the last multiple of 4 in the range and 20 is the first multiple of 4 in the range then total # of multiples of 4 in the range is (96-20)/4+1=20.

Or look at it in another way out of 20 consecutive even integers we have half will be even multiples of 4 and half will be even not multiples of 4.

Hope it's clear.
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Re: divisible by 12 Probability  [#permalink]

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shashankp27 wrote:
If integer C is randomly selected from 20 to 99, inclusive. What is the probability that $$C^3$$ - $$C$$ is divisible by 12 ?

a. 1/2
b. 2/3
c. 3/4
d. 4/5
e. 1/3

(C-1)C(C+1) should be divisible by 12.

Question is: How many of the integers from 20 to 99 are either ODD or Divisible by 4.

ODD=(99-21)/2+1=40
Divisible by 4= (96-20)/4+1=20
Total=99-20+1=80

P=Favorable/Total=(40+20)/80=60/80=3/4

Ans: "C"
##### General Discussion
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Re: divisible by 12 Probability  [#permalink]

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gaurav2k101 wrote:
How many of the integers from 20 to 99 are either ODD or Divisible by 4.
could you explain why have u take "either ODD"

Thank you

We know C^3-C=(C-1)C(C+1), product of 3 consecutive integers.

Now,
19,20,21
20,21,22
21,22,23
22,23,24
23,24,25
...
96,97,98
97,98,99
98,99,100

These are the entire set. Note; the middle value represents "C" and we know 20<=C<=99; (C-1) AND (C+1) are left and right values based on C.

Total count=99-20+1=80

We need to see how many of these sets will be divisible by 4;

If C=odd; left value=C-1=even; C+1=even; Multiplication of 2 Even numbers will always be divisible by 4 because it will contain at least two 2's in its factors.
e.g.
48,49,50
49=odd
48=even
50=even;
So, 48*49*50 must be a multiple of 4 as it has 2 even numbers. Thus, we count all odds C's, for it will make the left and right values even.

Or,
23, 24, 25
Since, 24 is a multiple of 4, it will also be divisible by 4. Thus, we count that too.

But, sets such as
21,22,23: will not be divisible by 4 as 21 AND 23 are odds and don't contain any 2 in their factors. and 22 is not divisible by 4.

forgot to mention:
the product of three consecutive integers will always be divisible by 3, so we don't need to mind that. We just need to make sure that there are at least 2 2's.
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Re: If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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C^3-C = C(C-1)(C+1)

Divisibility by 12 requires divisibility by both 4 and 3.

given C-1, C and C+1 are 3 consec. integers, divisibility by 3 is guaranteed. The question remains whether it is divisible by 4.

2 cases:
a) C is odd -> C-1 and C+1 are even which guarantees at least 2 factors of 2 and hence divisible by 4
b) C is even -> only C is even -> divisibility by 4 requires C to be divisible by 4

20-99 inclusive is 80 integers, 40 odd and 40 even. Of those that are even every other one is divisible by 4, that is half of the 40. Therefore there are 20 divisible by 4

Thus the probability is (40+20)/80 = 3/4
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Re: If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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Ah! I had to take some time to figure out this 1. This working might be helpful to you! C^3 - C= (c-1)(c)(c+1) --> this is some rule Any integer (c-1)(c)(c+1) will be divisible by 12, if any of these numbers are factors of 4 and 3, either alone or together.
Now, in a set of any 3 consecutive integers, one will automatically be divisible by 3. So the Q is simplied as whether the integer is a multiple of 4!

Now, there are 2 possibilities - C is even or odd.
a) C is even. In this case 2 numbers (c-1 and c+1) will be odd. This leads to 2 sub possibilities!
(i) C is a multiple of 4 - This means the integer will be divisible by 4. (We will have: odd X multiple of 4 X odd)
(ii) C is not a multiple of 4- This means the integer will NOT be divisible by 4. (We will have: odd X non multiple of 4 X odd)

b) C is odd. two numbers (c-1 and c+1) are divisible by 2 and therefore the set is divisble by 4.

Now the total numbers in the set are 80 (not 79, since 20 and 99 are included). The number of possibilities for C are
(a) (i) --> 99/4 - 19/4 = 24 -4 = 20
(a) (ii) --> 99/2 - 19/2 = 49 -9 = 40; reduce 20 from (a)(i) ==> 20
(b) --> 100/2 - 20/2 = 50 -10 = 40

Sum of (a) (i) + (a) (ii) + (b) --> 20+20+40 = 80 (80 is the total number of numbers between 20 and 99, this means we have considered all possibilities )

Favourable outcomes above (a) (i) + (b) = 60

Therefore probability = 60/80 or 3/4

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Re: If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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2
My approach was just two steps:
1) As we have (C-1)*C*(C+1) we can forget about devisibility by 3 and think only of div. by 4;
2) Then I didn't care about the total number of sets but considered that moving through number line and taking triples of sequent numbers we have only one option of four possible not to cover multiple of 4.
So, the answer must be 1-1/4 = 3/4.

I realize that it could be less straightforward if number of possible options was not 80, but in that case denominator of the answer would be smth less simple than 2,3,4,5 as we see in answers.
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If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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matvan wrote:
shashankp27 wrote:
If integer C is randomly selected from 20 to 99, inclusive. What is the probability that C^3 - C is divisible by 12 ?

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 1/3

My solution is longer, time consuming.
The given expression is (c-1) c (c+1), product of three consecutive integer and it is always divisible by 6. However, we have to find whether this is divisible by 12. we need an additional factor of 2.
I used the induction method. When c is 22, 26, 30 and so on up to 98, it is not possible.
There are 20 such values. Therefore, the answer 60/80, which is 3/4.
If anybody has a crisp solution please....

Where I say that induction method is the soul of Math's we should not forget that Induction methods with a little bit of Observation is the finest way to solve the question.

The key is "To question how and when will (c-1) c (c+1) be divisible by 12"

12 = 4*3

i.e. (c-1) c (c+1) will be divisible by 12 if it's divisible by 4 as well by 3

in order to make sure that this is divisible by 4

1) either (c-1) should be divisible by 2 so that (c+1) is also even and their product is divisible by 4 i.e. (c-1) can b any even number from 20 through 99 = (98/2)-(18/2) = 49-9 = 40 cases

2) or if (c-1) is odd then (c+1)will also be odd i.e. c must be the multiple of 4 itself i.e. c can be any multiple of 4 from 20 though 99 = (99/4)-(19/4) = 24-4 = 20 cases

Total favorable cases = 40 + 20 = 60

Hence, Probability = 60/80 = 3/4

I hope this helps!
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Re: If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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Bunuel : I am finding these type of question difficult to understand ..
Can you please simplify a bit more ..explaining how we determine the cases in which the given number is a multiple of the given divisor... please

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Re: If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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Quote:
If integer C is randomly selected from 20 to 99, inclusive. What is the probability that C^3 - C is divisible by 12 ?

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 1/3

We are given that an integer C is to be chosen at random from the integers 20 to 99 inclusive, and we need to determine the probability that C3 – C will be divisible by 12.

We should recall that when a number is divisible by 12, it is divisible by 4 and 3. We should also recognize that C3 – C = C(C2 – 1) = C(C – 1)(C + 1) = (C – 1)(C)(C + 1) is a product of three consecutive integers. Furthermore, we should recognize that any product of three consecutive integers is divisible by 3! = 6; thus, it’s divisible by 3. We have to make sure it’s also divisible by 4.

Case 1: C is odd

If C is odd, then both C – 1 and C + 1 will be even. Moreover, either C – 1 or C + 1 will be divisible by 4. Since C(C – 1)(C + 1) is already divisible by 3 and now we know it’s also divisible by 4, C(C – 1)(C + 1) will be divisible by 12.

Since there are 80 integers between 20 and 99 inclusive, and half of those integers are odd, there are 40 odd integers (i.e., 21, 23, 25, …, 99) from 20 to 99 inclusive. Thus, when C is odd, there are 40 instances in which C(C – 1)(C + 1) will be divisible by 12.

Case 2: C is even

If C is even, the both C – 1 and C + 1 will be odd. If C is a multiple of 4 but not a multiple of 3, then either C – 1 or C + 1 will be divisible by 3 (for example, if C = 28, then C – 1 = 27 is divisible by 3, and if C = 44, then C + 1 = 45 is divisible by 3). In this case, C(C – 1)(C + 1) will be divisible by 12.

If C is a multiple 4 and also a multiple of 3, then C is a multiple of 12 and of course C(C – 1)(C + 1) will be divisible by 12. Therefore, if C is even and a multiple of 4, then C(C – 1)(C + 1) will be divisible by 12. So, let’s determine the number of multiples of 4 between 20 and 99 inclusive.

Number of multiples of 4 = (96 – 20)/4 + 1 = 76/4 + 1 = 20. Thus, when C is even, there are 20 instances in which C(C – 1)(C + 1) will be divisible by 12.

In total, there are 40 + 20 = 60 outcomes in which C(C – 1)(C + 1) will be divisible by 12.

Thus, the probability that C(C – 1)(C + 1) will be divisible by 12 is: 60/80 = 6/8 = 3/4.

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Re: divisible by 12 Probability  [#permalink]

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Bunuel wrote:
If integer C is randomly selected from 20 to 99, inclusive. What is the probability that c^3-c is divisible by 12?
A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 1/3

Two things:
1. There are total of 80 integers from 20 to 99, inclusive: 20, 21, ..., 99.
2. C^3-C=(C-1)*C*(C+1): we have the product of 3 consecutive integers, which is always divisible by 3. So the question basically is whether (C-1)*C*(C+1) is divisible by 4.

Next, the only way the product NOT to be divisible by 4 is C to be even but not a multiple of 4, in this case we would have (C-1)*C*(C+1)=odd*(even not multiple of 4)*odd.

Now, out of first the 4 integers {20, 21, 22, 23} there is only 1 even not multiple of 4: 22. All following groups of 4 will also have only 1 even not multiple of 4 (for example in {24, 25, 26, 27} it's 26, and in {96, 97, 98, 99} it's 98, always 3rd in the group) and as our 80 integers are entirely built with such groups of 4 then the overall probability that C is not divisible by 4 is 1/4. Hence the probability that it is divisible by 4 is 1-1/4=3/4.

Hi Bunuel,
Product of 3 consecutive integers will be always NOT ONLY divisible by 3 BUT ALSO divisible by 2...Right?

Then,can we rephrase this question as whether (C-1)*C*(C+1) is divisible by 2?

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Re: If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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After watching the answer posted for this question, i would like to point out a method which requires least calculation and time.
Given expression -> (C-1)*C*(C+1) is always divisible by 3 (Product of three consecutive numbers)
Now the question here is to whether this will be divisible by 4 ?
This can be solved by using probability of selecting 3 numbers out of 4 numbers whose product is divisible by 4 where numbers are consecutive is 3/4
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If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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shashankp27 wrote:
If integer C is randomly selected from 20 to 99, inclusive. What is the probability that C^3 - C is divisible by 12 ?

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 1/3

Question: is What is the probability that$$\frac{C^3 - C}{12}$$ ---->$$\frac{C(C^2-1)}{4*3}$$ --->$$\frac{(C-1)C(C+1)}{4*3}$$?

(20,21,22,23),(24,25,26,27),(28,29,30,31),......,(96,97,98,99)

In above groups group of three numbers divisible by 12 and the group always excluded one number that is not divisible by 4.( any three consecutive numbers are always divisible by 3)

So the probability is $$\frac{3}{4}$$

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Re: If integer C is randomly selected from 20 to 99, inclusive.  [#permalink]

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Posting official solution of this problem.
Attachments official_4.PNG [ 164.44 KiB | Viewed 1586 times ]

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Re: divisible by 12 Probability  [#permalink]

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How many of the integers from 20 to 99 are either ODD or Divisible by 4.
could you explain why have u take "either ODD"

Thank you
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Re: divisible by 12 Probability  [#permalink]

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fluke wrote:
shashankp27 wrote:
If integer C is randomly selected from 20 to 99, inclusive. What is the probability that $$C^3$$ - $$C$$ is divisible by 12 ?

a. 1/2
b. 2/3
c. 3/4
d. 4/5
e. 1/3

(C-1)C(C+1) should be divisible by 12.

Question is: How many of the integers from 20 to 99 are either ODD or Divisible by 4.

ODD=(99-21)/2+1=40
Divisible by 4= (96-20)/4+1=20
Total=99-20+1=80

P=Favorable/Total=(40+20)/80=60/80=3/4

Ans: "C"

Would you please explain (1) The formulas you used for finding the # of ODD and "Divisible by 4" outcomes?
AND: (2) Why are we looking for the number of ODD integers within 20-99? I'm confused b/c 23 for example is not divisible by 12....

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Re: divisible by 12 Probability  [#permalink]

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Thank you very much for the explanation! It definitely clarified the concept Manager  Joined: 03 Dec 2012
Posts: 186
Re: If integer C is randomly selected from 20 to 99, inclusive  [#permalink]

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can v try and solve the problem in the foll. way: C^3-C = C(C^2-1)/12 so then v could check the probability for C/12. Now between 20 and 99 there are 7 integers i.e 24, 36, 48, 60, 72, 84, 96. I don't know how to proceed further.
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Re: divisible by 12 Probability  [#permalink]

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Bunuel wrote:
If integer C is randomly selected from 20 to 99, inclusive. What is the probability that c^3-c is divisible by 12?
A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 1/3

Two things:
1. There are total of 80 integers from 20 to 99, inclusive: 20, 21, ..., 99.
2. C^3-C=(C-1)*C*(C+1): we have the product of 3 consecutive integers, which is always divisible by 3. So the question basically is whether (C-1)*C*(C+1) is divisible by 4.

Next, the only way the product NOT to be divisible by 4 is C to be even but not a multiple of 4, in this case we would have (C-1)*C*(C+1)=odd*(even not multiple of 4)*odd.

Now, out of first the 4 integers {20, 21, 22, 23} there is only 1 even not multiple of 4: 22. All following groups of 4 will also have only 1 even not multiple of 4 (for example in {24, 25, 26, 27} it's 26, and in {96, 97, 98, 99} it's 98, always 3rd in the group) and as our 80 integers are entirely built with such groups of 4 then the overall probability that C is not divisible by 4 is 1/4. Hence the probability that it is divisible by 4 is 1-1/4=3/4.

nkhosh wrote:
Would you please explain (1) The formulas you used for finding the # of ODD and "Divisible by 4" outcomes?
AND: (2) Why are we looking for the number of ODD integers within 20-99? I'm confused b/c 23 for example is not divisible by 12....

Thank you for the great post 1. There are even # of consecutive integers in our range - 80 (from 20 to 99, inclusive). Out of even number of consecutive integers half is always odd and half is always even, thus numbers of odd integers in the given range is 40.

2. $$# \ of \ multiples \ of \ x \ in \ the \ range =$$
$$=\frac{Last \ multiple \ of \ x \ in \ the \ range \ - \ First \ multiple \ of \ x \ in \ the \ range}{x}+1$$.

So, for our case as 96 is the last multiple of 4 in the range and 20 is the first multiple of 4 in the range then total # of multiples of 4 in the range is (96-20)/4+1=20.

Or look at it in another way out of 20 consecutive even integers we have half will be even multiples of 4 and half will be even not multiples of 4.

Hope it's clear.

Hi Bunnel,

Is the below method correct:

As per the 30 sec approach: (c-1)c(c+1)

The initial 5 entries (before the multiple of 12) have 4 entries which have a multiple of 12
19.20.21
20.21.22
21.22.23
22.23.24
23.24.25

Then their are 8 groups of 12, and each having a probability of being divisible by 12 = 9/12=3/4
24.25.26 36.37.38
25 37
26
.
.
35.36.37 47.48.49..............8 such groups..

In the last 3 entries 2 are divisible by 12
96.97.98
97.98.99
98.99.100

So among the first 5 and last 3 - (4+2)/(5+3)=6/6=3/4

Thus the overall probability is 3/4.
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Re: divisible by 12 Probability  [#permalink]

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bagdbmba wrote:
Bunuel wrote:
If integer C is randomly selected from 20 to 99, inclusive. What is the probability that c^3-c is divisible by 12?
A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 1/3

Two things:
1. There are total of 80 integers from 20 to 99, inclusive: 20, 21, ..., 99.
2. C^3-C=(C-1)*C*(C+1): we have the product of 3 consecutive integers, which is always divisible by 3. So the question basically is whether (C-1)*C*(C+1) is divisible by 4.

Next, the only way the product NOT to be divisible by 4 is C to be even but not a multiple of 4, in this case we would have (C-1)*C*(C+1)=odd*(even not multiple of 4)*odd.

Now, out of first the 4 integers {20, 21, 22, 23} there is only 1 even not multiple of 4: 22. All following groups of 4 will also have only 1 even not multiple of 4 (for example in {24, 25, 26, 27} it's 26, and in {96, 97, 98, 99} it's 98, always 3rd in the group) and as our 80 integers are entirely built with such groups of 4 then the overall probability that C is not divisible by 4 is 1/4. Hence the probability that it is divisible by 4 is 1-1/4=3/4.

Hi Bunuel,
Product of 3 consecutive integers will be always NOT ONLY divisible by 3 BUT ALSO divisible by 2...Right?

Then,can we rephrase this question as whether (C-1)*C*(C+1) is divisible by 2?

I'm quite sure this works only with prime numbers. Otherwise you can prove divisibility of 2 by 2^N =) Re: divisible by 12 Probability   [#permalink] 13 Sep 2013, 04:43

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