Tarungaur wrote:
CrackVerbalHi, can we not substitute values in the equation (x-K) from the answer options and see which one is divisible by 3? If we get x-k to be divisible by 3 we shall get to the answer. Is there any other way to solve the question quickly?
Since its a
must be true type question and if you are using the answer options to solve it instead of seeing which one is divisible by 3 ,we should take each answer option and prove that in some particular case \( x(x – 1)(x – k)\) will not be divisible by 3. If you are using this approach, plugging in values for x can save a lot of time.
#Approach 1: Plugging in values for x.
Here we need to find a case where x(x – 1)(x – k) is not divisible by 3.
Lets take 3 consecutive numbers: 9,10,11
Lets start by assuming x =11 as x and x-1 is not a multiple of 3
x(x – 1)(x – k) = 11*10 *(11 - k) .
Since 11 and 10 are not divisible by 3, if (11- k) is also not divisible by 3 ,then we will get a scenario where x(x – 1)(x – k) is not divisible by 3.
Check each answer option and see for which one , 11-k is not divisible by 3.
A. -4 => (11--4) =15 is divisible by 3
B. -2 => (11--2) = 13 is not divisible by 3. That means 11*10*13 is not divisible by 3.
Since its a must be true question we can say that when k=-2 ,x(x – 1)(x – k) is not always divisible by 3.
Option B is the correct answer.
C. -1 => (11--1) =12 is divisible by 3
D. 2 => (11-2) =9 is divisible by 3
E. 5 => (11-5) =6 is divisible by 3
#Approach 2:
We know that in every 3 consecutive numbers, there will be a multiple of 3.
Here it's given that x(x – 1)(x – k) must be evenly divisible by 3 and we need to find a value of k where x(x – 1)(x – k) is not always divisible by 3.
Let's say x,x-1,x-2 are 3 consecutive terms.
Case 1: if x is a multiple of 3, x(x – 1)(x – k) is always be divisible by 3 , irrespective of any values of k
Case 2: x is not a multiple of 3 and x-1 is multiple of 3.
Here also, x(x – 1)(x – k) is always be divisible by 3 , irrespective of any values of k
Case 3: x and x-1 is not a multiple of 3, That means x-2 should be a multiple of 3. As we know in every 3 consecutive numbers, there will be a multiple of 3.
If x-2 is a multiple of 3, then if we add or subtract 3 to x-2 each time, it should be also a multiple of 3.
So we can conclude that x-2, x-5, x-8, x-11... as well as x-2,x+1,x+4, x+7 will also be multiple of 3.
Hence we can conclude that value of K should be either 2,5,8,11.. or 2,-1,-4,-7.
Combining all the possible values of K in which x(x – 1)(x – k) must be divisible by 3 will be in an Arithmetic series ...-7,-4,-1,2,5,8.. with a common
difference 3.
If you analyse the answer options, Option B. -2 is not in list .Therefore, we can conclude that if k=-2 , we cannot say that x(x – 1)(x – k) must be evenly divisible by 3
#Approach 3: Spot the pattern in the answer options. Here in each answer options there is a common difference of 3 except Option B. i.e. -2. If you can apply the Arithmetic series logic explained in Approach 2 , we can easily figure that Option B would be the answer.
I hope this clears all your doubts.
Let me know in case if you need any further help on this.
Thanks,
Clifin J Francis,
GMAT SME _________________