Carcass wrote:
What is the smallest integer that is divisible by 2, 3, 4, 5, 6, 8, and 9?
A) 180
B) 270
C) 360
D) 540
E) 720
One approach is to
test each answer choice (starting with the smallest answer choice) to find the smallest value that's divisible by 2, 3, 4, 5, 6, 8, and 9
Another option is to
BUILD the number in question. Let's build the number...
-----ASIDE---------------------
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 NConsider 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)
-----ONTO THE QUESTION!---------------------
Let's say that N = the smallest integer that is divisible by 2, 3, 4, 5, 6, 8, and 9
So if N is divisible by 2, we know that a
2 must be hiding in its prime factorization. That is N = (
2)(?)(?)(?)(?)(?)
Likewise, if N is divisible by 3, there must be a
3 hiding in the prime factorization of N. That is N = (2)(
3)(?)(?)(?)(?)
Likewise, if N is divisible by 4, there must be two 2's hiding in the prime factorization of N. We already have one 2, so we need to add one more 2 to get: N = (2)(3)(
2)(?)(?)(?)
Since N is divisible by 5, we need to add a
5 to get: N = (2)(3)(2)(
5)(?)(?)
In order to be divisible by 6, there must be one 2 and one 3 in the prime factorization of N. Since we already have one 2 and one 3, we can be certain that N is already divisible by 6
To be divisible by 8, there must be three 2's hiding in the prime factorization of N. We already have two 2's, so we need to add one more 2 to get: N = (2)(3)(2)(5)(
2)(?)(?)
Finally, to be divisible by 9, there must be two 3's hiding in the prime factorization of N. We already have one 3, so we need to add one more 3 to get: N = (2)(3)(2)(5)(2)(
3)
So, N = (2)(3)(2)(5)(2)(3) = 360
Answer: C