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# If x is an integer and 1 ≤ x ≤ 100, what is the probability that x(x +

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Math Expert
Joined: 02 Sep 2009
Posts: 60687
If x is an integer and 1 ≤ x ≤ 100, what is the probability that x(x +  [#permalink]

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01 Nov 2019, 06:59
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Difficulty:

95% (hard)

Question Stats:

31% (03:37) correct 69% (03:29) wrong based on 42 sessions

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If x is an integer and $$1 ≤ x ≤ 100$$, what is the probability that $$x(x + 1)$$ is a multiple of 12 and 9?

(A) 1/10
(B) 1/20
(C) 2/25
(D) 11/100
(E) 17/100

Are You Up For the Challenge: 700 Level Questions

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Joined: 20 Jul 2017
Posts: 1265
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Re: If x is an integer and 1 ≤ x ≤ 100, what is the probability that x(x +  [#permalink]

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01 Nov 2019, 07:48
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Bunuel wrote:
If x is an integer and $$1 ≤ x ≤ 100$$, what is the probability that $$x(x + 1)$$ is a multiple of 12 and 9?

(A) 1/10
(B) 1/20
(C) 2/25
(D) 11/100
(E) 17/100

Are You Up For the Challenge: 700 Level Questions

To be a multiple of 12 & 9 --> to be a multiple of LCM(12, 9) = 36
--> We have to find that x(x+1) is a multiple of 36

Total = 100
Note that, one of x or (x+1) should be a multiple of 9 and the other should be a multiple of 4 or one should be a multiple of 36
--> Favorable values = {8*9, 27*28, 35*36, 36*37, 44*45, 63*64, 71*72, 72*73, 80*81, 99*100} = 10

Required Probability = 10/100 = 1/10

IMO Option A
Math Expert
Joined: 02 Aug 2009
Posts: 8332
Re: If x is an integer and 1 ≤ x ≤ 100, what is the probability that x(x +  [#permalink]

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01 Nov 2019, 19:14
Bunuel wrote:
If x is an integer and $$1 ≤ x ≤ 100$$, what is the probability that $$x(x + 1)$$ is a multiple of 12 and 9?

(A) 1/10
(B) 1/20
(C) 2/25
(D) 11/100
(E) 17/100

Are You Up For the Challenge: 700 Level Questions

$$x(x + 1)$$ is a multiple of 12 and 9 MEANS multiple of LCM(12,9) or 36
x and x+1 are consecutive numbers, so at the maximum, only one one will be a multiple of 3.
So we are looking at x or x+1 being divisible by 9, and then check if the even number in x and x+1 is divisible by 4
(I) Odd multiple of 9
Whenever there is an odd multiple of 9, one of the even digits on two sides will be divisible by 2 and the other by 4. Hence all odd multiple of 9 will qualify..
9, 27, 45, 63, 81, 99- 6 ways
(II) Even multiple of 9
(a) Whenever the even multiple is just by 2..NO, so eliminate 9*2, 9*6
(b) Whenever the even multiple is by 4..the number itself is completely divisible by 36, so we get two values where x or x+1 is divisible by 9.. for example 35*36 or 36*37.
so 9*4, 9*8 will give TWO each - 4 ways

Total -- 6+4=10..
Possible values of x = 1 to 100

Probability -- $$\frac{10}{100}=\frac{1}{10}$$

A
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Re: If x is an integer and 1 ≤ x ≤ 100, what is the probability that x(x +   [#permalink] 01 Nov 2019, 19:14
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