Vincen wrote:What is the positive integer n?
(1) For every positive integer m, the product m(m + 1)(m + 2) ... (m + n) is divisible by 16
(2) n^2 - 9n + 20 = 0
The OA is C.
I can not see how to use statement (1). Experts, could you help me?
Statement 2 seems relatively easier than Statement 1 to handle, so let's start with it. Moreover, it may also give us a hint on how to approach Statement 1.
Statement 2: n^2 - 9n + 20 = 0
=> n = 4 or 5. No unique value. Insufficient.
Statement 1: For every positive integer m, the product m(m + 1)(m + 2) ... (m + n) is divisible by 16.
Let's understand the meaning of the statement. It means that for
every positive integer m, the product m(m + 1)(m + 2) ... (m + n) is divisible by 16. The statement is true for all the possible values of m such as 1, 2, 3, 4, ...
Say m = 1, then if n = say 5, then m(m + 1)(m + 2) ... (m + n) = 1*2*3*4*5*6. It is divisble by 16.
Say m = 2, then if n = say 4, then m(m + 1)(m + 2) ... (m + n) = 2*3*4*5*6. It is also divisble by 16.
We do not get the unique value of n. Insufficient.
(1) and (2):
Case 1: Say m = 2, and n = 4, then m(m + 1)(m + 2) ... (m + n) = 2*3*4*5*6. It is divisible by 16.
Case 2: Say m = 1, and n = 4, then m(m + 1)(m + 2) ... (m + n) = 1*2*3*4*5. It is NOT divisible by 16, so this is not a valid case. Statement 2 states that For
every positive integer m, the product m(m + 1)(m + 2) ... (m + n) is divisible by 16.
So, n ≠4.
Case 3: Say m = 1, and n = 5, then m(m + 1)(m + 2) ... (m + n) = 1*2*3*4*5*6. It is divisible by 16. Thus, n = 5. Sufficient.
Though there is no need to check at m = 2, since this is going to give us one 2. Let's do it for the sake of understanding.
Case 4: Say m = 2, and n = 5, then m(m + 1)(m + 2) ... (m + n) = 2*3*4*5*6*7. It is divisible by 16.
The correct answer:
C
Hope this helps!
-Jay
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