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If M and N are positive integers, is (MN)/15 an integer

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Bunuel
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If M and N are positive integers, is (MN)/15 an integer [#permalink] New post   13 Mar 2020, 01:48
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If M and N are positive integers, is (MN)/15 an integer

(1) The least common multiple of M and N is 80
(2) The greatest common factor of M and N is 4


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Re: If M and N are positive integers, is (MN)/15 an integer [#permalink] New post   13 Mar 2020, 05:04
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Bunuel wrote:
If M and N are positive integers, is (MN)/15 an integer

(1) The least common multiple of M and N is 80
(2) The greatest common factor of M and N is 4



Question: Is MN/15 an Integer?

For M*N to be an integer, M and N must have a factor 3 and.5 both

Statement 1: The least common multiple of M and N is 80

I.e. M and N have no prime factor 3 I.e. MN is NOT divisible by 15

SUFFICIENT

Statement 2: The greatest common factor of M and N is 4

But one M and N may have a Prime factor 3 and other 5 hence MN may or may NOT be divisible by 15

Case 1: M = 4 and N = 60 NO
Case 2: M = 12 and N = 20 YES

NOT SUFFICIENT

Answer: Option A
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Re: If M and N are positive integers, is (MN)/15 an integer [#permalink] New post   01 Jan 2021, 16:31
For statement 2 both cases are divisible by 15. Do you have another example?
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Re: If M and N are positive integers, is (MN)/15 an integer [#permalink] New post   11 May 2021, 06:06
Bunuel wrote:
If M and N are positive integers, is (MN)/15 an integer

(1) The least common multiple of M and N is 80
(2) The greatest common factor of M and N is 4



can't stress it enough how important prime factorisation is for questions like these:-

Ques-Stem
is \(MN/15\) = Integer ? or -----> \(MN/3*5\)= integer ?

St1> LCM of M & N is 80. okay, let's prime factorise 80, the aim behind this is that we are trying to find out M and N consists of what all prime numbers? PF. of \(80= 2^4*5\)

So we if try and visualise what MN/15 will look like, it will look something like this= \(2^4*5/3*5\)

clearly, we're short of a 3 in the numerator. this means this value can't be an integer because MN is missing a prime factor, which is 3

Stmt2> HCF of M and N is 4, now HCF does not go a very long way, mainly because an HCF is just COMMON PRIME FACTORS IN THEIR HIGHEST POWER. what that means is:- we don't know if it only consists of a 2 or 2 with one three or maybe a 2, a 3 or a 5. HCF of 4 only means 2^2 was the highest power between the prime factors of M and N, maybe there were other factors too, we will never know. clearly insufficient.



PS: when in doubt, remember, \(HCF of (M,N) * LCM (M,N)= M*N\). so, if you need to confirm your answer is right. multiply the HCF and LCM, you will get 3200 for the product of M and N. and \(3200/15\) is not an integer, so the conclusion we arrived via stmt 1 is correct. ;)
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Re: If M and N are positive integers, is (MN)/15 an integer [#permalink] New post   15 May 2021, 07:01
sejalmali wrote:
For statement 2 both cases are divisible by 15. Do you have another example?


sejalmali, not sure if you still need it but M=4 and N=8 is one.
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Re: If M and N are positive integers, is (MN)/15 an integer [#permalink] New post   04 Jun 2023, 04:49
Video solution by TTP: https://youtu.be/u64h6J3H9H0?t=3047
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Re: If M and N are positive integers, is (MN)/15 an integer [#permalink]
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