What is the positive integer n?

(1) For every integer m, the product m(m + 1)(m + 2) ... (m + n) is divisible by 16

(2) n^2 - 9n + 20 = 0

OPEN DISCUSSION OF THIS QUESTION IS HERE: what-is-the-positive-integer-n-1-for-every-positive-126636.html

St2:
n = 4 or 5. insufficient.

st1:
the product is divisible by 16. say the product is x, then we can write x/16 = 2/2^4. so as long as x has four 2's, we can cancel out 16. the statement says this works for every integer m, so let's say m=1. if m = 5, then x =720 which is divisible by 16. if m = 4, then x = 120, which is not divisible by 16. if m = 7, then x = 2520 which is also divisible by 16. Insufficient.

using st2 and st1, we know n is 5. Sufficient.

Ans C

Got E.

I used the same method as Fistail:

If m=1, then n > 5
If m=2, then n > 4
and so on...

So (1) is insufficient.

For (2), n = 4 or 5. So (2) is insufficient as well.

(1) and (2) doesn't provide if n is 4 or 5 so E.

Is there a way to do this problem faster?

if m=2, m(m+1)(m+2)(m+3)(m+4) =(2x3x4x5x6) is divided by 16.

so n here is 4.

nope. in this case n = 4, in some other cases n=5 or could be more or less.

if m = 1, n = 5.
if m = 8, n = 3
if m = 15, n =2
if m = 16, n =1.

the divisibility of the expression m(m+1)(m+2)(m+3)(m+4) by depends upon n.

therefore it is E.

imo, this is really good question.

C for me.
16= 2*2*2*2
(1) If we know that there are at least 4 even numbers, then you know that it will be divisible by 16. However, you don't have to have 4 even numbers because if m=2, then next even term is m=4. The next even term after that is m=6. This will only require 3 even terms to satisfy the equation; thus, if m is odd, n=6 will satisfy the equation. If m is even, n=4 will satisfy the equation. In sum, n=4 or n=6
INSUFFICIENT.

(2) n=5 or n=4. INSUFFICIENT.

Together, we know that n=4
SUFFICIENT

Here's my attempt at it:

1.

this doesn't tell you anything specific about n. plug in 1 for m (since 1 is simple and M can be any integer) and see what works.

1(2) = 2
1(2)(3) = 6
1(2)(3)(4) = 24
1(2)(3)(4)(5) = 120
1(2)(3)(4)(5)(6) = 720

720 is divisible by 16, which means n = 5 works. this is just one example of an N that works, so you can't be sure that 5 is it, but you can be sure that 1, 2, 3 and 4 do not work (since 2, 6, 24, 120 are not divisible by 16)

as you can see, this is the same as (n+1)! = 16n. We know that 5 works, but there are tons of other possibilities greater than 5 that could be divisible by 16. Since it's DS and not PS we don't need to find them, just know that they exist and that A is NOT SUFFICIENT.

2.

break this down:

(n-4)(n-5) = 0

this means that N is = to either 4 or 5. obviously this isn't sufficient on it's own since we get two possible answers. NOT SUFFICIENT

but put them together and see what happens!

we already know that 4 does not work from the first statement, but 5 does. this gives 5 as the only possible answer and we get C is SUFFICIENT.

now if you went through statement 1 without testing for possibilities you could see from statement 2 that the answer is 4 or 5. then just go back to statement one and plug in 4 and 5 and see if one of them works.

m = 1 n = 4

1(1+1)(1+2)(1+3)(1+4)
1(2)(3)(4)(5) = 120
120 is not divisible by 16.

m = 1 n = 5

1(1+1)(1+2)(1+3)(1+4)(1+5)
1(2)(3)(4)(5)(6) = 720
720/16 = 46

so using both statements we see that the only possible answer is 5. C is sufficient!

Why are you guys not considering the vaule of m negative....

M is juss a interger.It can be negative and zero also

The integers (Latin, integer, literally, "untouched," whole, entire, i.e. a whole number) are the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero.

because the stem tells us that N is positive.
and Statement 1 tells us that "for every integer m, the product..."

so we're looking for a positive n and we can plug any integer we want into the equation. plugging in -1, -2, -3 etc will just give us 0 for the answer as long as n > m. -1(-1 +1) = -1(0), and so on

and since 0 is divisible by everything it will work for 16

I agree with Fistail here and my pick is E.
According to 1st statement M can be any integer thus there is no fiexed value for n.
from 2. n can be 4 or 5.

Thus it has to be E. Unless i am missing some logic in statement 1 but i cannot see any way where we can assign a fixed value for m or n.
I think this is one of 700+ questions where you tend to pick C because you know it is a high score question.