The LCM of two numbers is divisible by the highest single digit ......

Question

The LCM of two distinct positive integers is divisible by the highest single digit power of 3, while their HCF is 12. Which of the following could be the smaller number between the two?
 
  A.  6
 B.  24
 C.  36
 D.  300
 E.  360

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ArunSharma12 · Accepted Answer

LCM = 3^9*k 
HCF = 2^2*3 
smaller number must be multiple of 12 and have 1 as the power of 3 .

24 = 2^3*3 
 36 = 2^2*3^2 
 300 = 2^2*3*5^2 
 360 = 2^3*3^2*5 
smaller number can be 24.
Ans: B

arvind910619 · Answer

Hi experts,

I have a doubt regarding this question.
I was able to eliminate A,C and E but not B and D.
Both B and D both contain prime factorization of HCF = 2^2*3 
Both B and D can be the answer. 
Can you please help me out in eliminating D. What mistake I am committing on this question?

rvgmat12 · Answer

HCF is 12 = 2^2*3

So the smaller number cannot have more than 3^1.. hence D is incorrect

Kinshook · Answer

Given: The LCM of two distinct positive integers is divisible by the highest single digit power of 3, while their HCF is 12.

Asked: Which of the following could be the smaller number between the two?
 
Highest single digit power of 3 = 9
Let LCM be 9k
HCF = 12

Product of numbers = 9k*12 = 12*12m
3k = 4m
k = 4; m = 3
36*12 = 12*12*3
If k = 8; m = 6
9*8*12 = (12*2)*(12*3)

IMO B

Can,Will · Answer

gmatophobia   KarishmaB  
please provide a detailed solution

Can,Will · Answer

Product of numbers = 9k*12 = 12*12m
can you elaborate above step?

gmatophobia · Answer

Can,Will Here is how I approached this question - The LCM of two distinct positive integers is divisible by the highest single digit power of 3 Inference : We have two numbers (let's assume them A and B) Highest single digit power of 3 = 3^9 → As 9 is the highest single-digit number LCM(A, B) is divisible by 3^9 → This statement also tells us that between A and B, at least one of the numbers is divisible by 3^9 Let's draw two prime boxes on our scratchpad. These are the prime boxes for A and B. Keep a mental note ⇒ we must have at least the ninth power of 3 in one of the boxes. At this stage, we do not know in which box that goes, but at least one of the boxes has it. Step 1.png while their HCF is 12 Inference : 12 = 2^2 * 3 → As 12 is the HCF, each box must contain 2^2 * 3 Again a mental note ⇒ A and B both do not contain any other common prime numbers or any other powers of 2 and 3. Let's put this information into our scratchpad Step 2.png Which of the following could be the smaller number between the two? Note ⇒ It's a 'could be' type of question. Hence, we can have multiple possibilities overall. We need to find the smaller number between the two (numbers A & B). Let's look at the options (as we need to find the smallest number, always start from the smaller one) A. 6 Nah! We can't have 6 as the smaller number. We know that the minimum value of A and B is 12. B. 24 Well, this could be possible. Suppose we have an extra 2 in one of the prime boxes and 3^8 in the other prime box, the constraints set in the question are not violated. Hence, 24 could be the smallest number. ACE - 2.png I wouldn't check further. Time is precious

; mark B, and move on! Option B

Can,Will · Answer

Thank you for sharing such a detailed explanation. much appreciated

KarishmaB · Answer

The wording of the question is not good and as written, the question makes no sense to me - until and unless the author intended to write something else. As written, both (B) and (D) are equally valid answers. What is meant by "divisible by the highest single digit power of 3" is something I will only guess. The two numbers will be 12*a and 12*b where a and b are co-prime. Also, one of a and b has 3^8 as a factor (since a 3 is already there in 12) but the other will have no more 3s. Which is the smaller number, I can't say. It is certainly possible to have the following cases: 12 ext{ and } 12*3^8 so the smaller number here is the first one. 12*5^{26} ext{ and } 12*3^8 so the smaller number here is the second one. Still I ignore this and move forward assuming the author wants to check about the first number. It can easily be 24 (=12*2) and can be 300 (= 12 * 25) The two numbers could be 24 ext{ and } 12*3^8 - HCF is 12 and LCM has 3^9 as factor 300 ext{ and } 12*3^8 - HCF is 12 and LCM has 3^9 as factor Answer Indeterminate LCM and GCF is tricky concept. Check this post for some properties of the two and how to handle them: https://anaprep.com/number-properties-s ... roperties/

bumpbot · Answer

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The LCM of two numbers is divisible by the highest single digit ......

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