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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.

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

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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

(1) tells us that n is at least 5, as the product of 6 or more consecutive integers is always a multiple of 16. The product of fewer than 6 consecutive integers need not be a multiple of 16.
NOT SUFF
(2) n is either 4 or 5
NOT SUFF
(T) n=5
SUFF

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

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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

(1) tells us that n is at least 5, as the product of 6 or more consecutive integers is always a multiple of 16. The product of fewer than 6 consecutive integers need not be a multiple of 16. NOT SUFF (2) n is either 4 or 5 NOT SUFF (T) n=5 SUFF

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

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

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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

I think it's 'A'. n=15 always holds good, it's not really asking for a minimum value of 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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the 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.

From 1st alone we cant say. n can has so many values depends on m.
From 2nd option we got n 4 and 5...

2nd is also insufficient...

When we put both value of n in 1st.It satisfied quation on diff value of m not every value of m.

We are trying only +ve value of m.. m can be -ve also.When we put m as -1,-2,-3,-4 and n as 4 or 5.we get the product m(m + 1)(m + 2) ... (m + n) =0 which is not divisible by 16.

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.

Quote:

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

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.

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

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

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

(1) tells us that n is at least 5, as the product of 6 or more consecutive integers is always a multiple of 16. The product of fewer than 6 consecutive integers need not be a multiple of 16. NOT SUFF (2) n is either 4 or 5 NOT SUFF (T) n=5 SUFF

I got the same answer, but n can be less than 5 for (1). For example, say m=2 and n=4. You have: 2*3*4*5*6, which satisfy the multiplication.

In fact, if m is even, you only need n=4. This is true for all cases. Try it out.

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.

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.

I consider m negative and zero. I break it down to even and odd, which consider all.

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

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.

I consider m negative and zero. I break it down to even and odd, which consider all.

But when m = 0 the product m(m + 1)(m + 2) ... (m + n) =0 which is not divisible by 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.

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.

(1) For every integer m, m(m+1)...(m+n) is a multiple of 16 for EVERY integer m if and only if n>4. (1) tells us that n is at least 5. NOT SUFF
(2) n is either 4 or 5 NOT SUFF
(T) n=5 SUFF

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