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

Question

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

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CareerGeek · Accepted Answer

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

chetan2u · Answer

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

MissionAdmit · Answer

There are only four values of x for which x (x+1) is a multiple of 36 and these are 35,36,71 and 72. All these 10 values are not multiples of 12 AND 9. AND not OR. Confused by the answer here

sanjitscorps18 · Answer

Let's say we take the number 27*28 as x(x+1)
Now the question states that 27*28 should be a multiple of 9 and 12. 
So 
27*28 divided by 12 gives 63
27*28 divided by 9 gives 84

Hence 27*28 and the other numbers are divisible by 12 and 9 both as a multiple of x(x+1) together.

Dereno · Answer

Multiples of 4 and 9 less than 100 are given below.

Multiples of 4 = 4, 8 ,12,16,20,24, 28 ,32, 36,40, 44 ,48,52,56,60, 64 ,68,72,76, 80 , 84, 88,92,96, 100

Multiples of 9 =  9 ,18, 27 ,36, 45 ,54, 63 ,72, 81 ,90, 99 .

Let’s look for consecutive numbers (8,9) , (27,28) , (44,45) , (63,64) , (80,81) and (99,100)

We know that x(x+1) is a multiple of 36 . Either x is 36 or x+1 is 36.

We get, (35,36) or (36,37). Another multiple of 36 is 72. We get, (71,72) or (72,73).

Total count is 10.

Probability = 10/100 =  1/10

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

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GMAT Problem Solving (PS) Questions

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

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