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LCM of 1 to 8 numbers should be a three digit number (= 840)

All 4 digit OA can be discarded

Answer = A
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I am confused!! Can someone please explain it to me in a more basic language?
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gmatgambler
If K is the least positive integer that is divisible by every integer from 1 to 8 inclusive, then K =

A. 840
B. 2,520
C. 6,720
D. 20,160
E. 40,320

Integer that is divisible by every integer from 1 to 8 inclusive is : 8!

\(8! = 8*7*6*5*4*3*2\)

Check the options to find the least value which is divisible by 2,3,5 & 7

Only option (A) 840 , matches as it is divisible by 2,3,5 & 7

So, Correct answer must be (A)
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gmatgambler
If K is the least positive integer that is divisible by every integer from 1 to 8 inclusive, then K =

A. 840
B. 2,520
C. 6,720
D. 20,160
E. 40,320

We need to determine the LCM of 2, 3, 4, 5, 6, 7, and 8. Factoring each number into primes, we have:

2, 3, 2^2, 5, 2 x 3, 7, 2^3

So the LCM is 2^3 x 3 x 5 x 7 = 840.

Answer: A
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gmatgambler
If K is the least positive integer that is divisible by every integer from 1 to 8 inclusive, then K =

A. 840
B. 2,520
C. 6,720
D. 20,160
E. 40,320

A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)

K is divisible by every integer from 1 to 8 inclusive
This means that there's a 2 "hiding" within the prime factorization of K, a 3 "hiding" within the prime factorization of K, a 4 "hiding" within the prime factorization of K, etc.

So, let's begin with a 2 "hiding" within the prime factorization of K.
This means that K = (2)(other numbers)

Also, if there's a 3 "hiding" within the prime factorization of K, then we need to add a 3 like so: K = (2)(3)

There's a 4 "hiding" within the prime factorization of K.
Since 4 = (2)(2), then we need to add a SECOND 2 to get: K = (2)(2)(3)

There's a 5 "hiding" within the prime factorization of K, so we'll add a 5 to get: K = (2)(2)(3)(5)

There's a 6 "hiding" within the prime factorization of K.
Since 6 = (2)(3), we can see that we ALREADY have a 6 "hiding" in the prime factorization: K = (2)(2)(3)(5)

There's a 7 "hiding" within the prime factorization of K, so we'll add a 7 to get: K = (2)(2)(3)(5)(7)

There's an 8 "hiding" within the prime factorization of K.
Since 8 = (2)(2)(2), we need to add a THIRD 2 to get: K = (2)(2)(2)(3)(5)(7)

We have now ensured that K is divisible by every integer from 1 to 8 inclusive. This means that we have found the LEAST possible value of K that satisfies the given conditions.
So, K = (2)(2)(2)(3)(5)(7) = 840

Answer: A

Cheers,
Brent
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