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

Are You Up For the Challenge: 700 Level Questions

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

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards