If the least common multiple of the first 800 positive integers is n

Question

If the least common multiple of the first 800 positive integers is n, then what is the least common multiple of the first 801 positive integers in terms of n ?

A. n
B. 9n
C. 89n
D. 800n
E. 801n

BrentGMATPrepNow · Accepted Answer

The least common multiple of the first 800 positive integers is n  
So, n is a multiple of 1, a multiple of 2, a multiple of 3, a multiple of 4, . . . a multiple of  9 , . . . ., a multiple of  89 , . . . a multiple of 799, and a multiple of 800

What is the least common multiple of the first 801 positive integers in terms of n ?  
801 = ( 9 )( 89 )
In other words, 801 is a multiple of  9 , and 801 is a multiple of  89 
Since n is already a multiple of  9  and a multiple of  89 , we know that n must be a multiple of 801

So, if n is the least common multiple of the first 800 positive integers, n is also the least common multiple of the first 801 positive integers.

Answer: A

Cheers,
Brent

Kezia9 · Answer

Hi Brent,

I solved it as follows. Please correct me if I am wrong.

I assumed the numbers to be 1,2,3,4,5. LCM is 60
LCM of 1,2,3,4,5,6 is also 60.

Hence A.

BrentGMATPrepNow · Answer

It's a useful strategy to examine a smaller case (first 5 positive integers vs the first 800 positive integers), but it happens to be a lucky coincidence that you found the correct answer. 
For example, if you had examined the first 4 positive integers, you would have gotten a different answer. 
If the numbers are 1, 2, 3, 4, then the LCM = 12
But the LCM of 1, 2, 3, 4, and 5 is not 12 (it's 60)

Cheers, 
Brent

Kinshook · Answer

Asked: If the least common multiple of the first 800 positive integers is n, then what is the least common multiple of the first 801 positive integers in terms of n ?

801 = 9*89 = 3^2*89

Since n is already a multiple of 9 and a multiple of 89.

IMO A

BrushMyQuant · Answer

Given that LCM of the first 800 positive integers is n and we need to find what is the least common multiple of the first 801 positive integers in terms of n

801 = 9 * 89
and both 9 and 89 are already present in the numbers from 1 to 800

=>  LCM(First 800 numbers) = LCM(First 801 numbers)  = n

Example: LCM of First 5 positive integers (1, 2, 3, 4, 5) = 3 * 4 * 5 = 60
LCM of First 6 positive integers (1, 2, 3, 4, 5, 6) is also 60 as 6 = 2*3 and both 2 and 3 are already considered in numbers from 1 to 5

So,  Answer will be A 
Hope it helps!

To learn more about LCM and GCD watch the following videos

https://www.youtube.com/watch?v=7jPV6JD0908

https://www.youtube.com/watch?v=vtSsAjcF-0Y

bumpbot · Answer

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If the least common multiple of the first 800 positive integers is n

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