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01 Nov 2019, 06:59
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
30% (02:59) correct
70%
(03:16)
wrong
based on 445
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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
(A) 1/10
(B) 1/20
(C) 2/25
(D) 11/100
(E) 17/100
Are You Up For the Challenge: 700 Level Questions
01 Nov 2019, 07:48
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Bunuel
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
01 Nov 2019, 19:14
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Bunuel
\(x(x + 1)\) 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
