Problem Solving

Hi,

The answer to this is E) but could not figure out why. Can anyone please explain? Thank yoU!

19. The product of the first twelve positive integers is divisible by all of the following EXCEPT
(A) 210
(B) 88
(C) 75
(D) 60
(E) 34

You need to split the number into factors here ...

210 = 2 * 3 * 5 * 7
88 = 8 * 11
75 = 3 * 5 * 5
60 = 5 * 12
34 = 2 * 17

Since we are dealing with the first 12 positive numbers, 17 can't be in
it. Hence 34

Dunkin,

I enjoy the math problems you are posting up. It’s a good way to practice.

In some cases, to save time, I’m trying to solve the questions backwards.


I see that you give away the answer in the first line. Could you please post the answer in the last line after the problem? Or you don’t even have to post the answer, which is even better. I’m sure some math wiz will come up with it.


Thanks.

NS88

210 = 21*10 = 7*3*5*2 (Hence possible)
88 = 2*2 (or 4)*2*11 (Again possible)
75 = 5*5*3 (Again possible)
60 = 3*2*5*2 (Again possible)
34 = 17*2 (Not possible. 17 is a prime number and it can't be broken down any further)

The product of the first twelve positive integers is:
1*2*3*4*5*6*7*8*9*10*11*12=12!
For a number to be exactly divisible by 12! it must be a factor.
Considering the options given;
-210
210=3*7*10
3, 7 and 10 can be factored out from 12! without a fractional remainder.
Therefore,210 is a factor.
-88=11*8
11 and 8 can be factored from 12! Thus, 88 is a factor
75 is a factor of 12!
-60 =12*5
12 and 5 can also be factored so 60 is a factor
-34=2*17
2 can be factored but 17 cannot be factored out.
Therefore,34=2*17 cannot be divided by 12!