das.ashmita 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
OA in the source is C
But I think its E
Statement 1: For every positive integer m, the product m(m + 1)(m + 2) ... (m + n) is divisible by 16
Since m is positive, m(m+1)(m+2)...(m+n) represents the product of CONSECUTIVE positive integers.
Every other consecutive EVEN integer is a MULTIPLE OF 4.
Thus, if the product includes 3 consecutive EVEN factors, there are two cases:
Case 1: (multiple of 2)(multiple of 4)(multiple of 2)
This product must be divisible by (2)(4)(2) = 16.
Case 2: (multiple of 4)(multiple of 2)(multiple of 4)
This product must be divisible by (4)(2)(4) = 32.
In each case, the product is divisible by 16.
Thus, if the product includes at least 3 consecutive even factors, it will be divisible by 16.
To GUARANTEE that the product will include at least 3 consecutive even factors, the MINIMUM value of n is 5:
m(m+1)(m+2)(m+3)(m+4)(m+5) = the product of 6 consecutive integers.
This product will be composed of exactly 3 consecutive odd factors and exactly 3 consecutive even factors.
If n=4 -- implying a total of 5 consecutive factors -- then the product could be (odd)(even)(odd)(even)(odd).
In this case, the product will not include at least 3 consecutive even factors, with the result that it might not be divisible by 16.
To illustrate:
Neither 1*2*3*4*5 nor 17*18*19*20*21 is divisible by 16.
Since the product must include at least 3 consecutive even factors to GUARANTEE that it will be divisible by 16, n≥5.
No way to determine the exact value of n.
INSUFFICIENT.
Statement 2: n² - 9n + 20 = 0
(n-4)(n-5) = 0.
Since it's possible that n=4 or n=5, INSUFFICIENT.
Statements 1 and 2 combined:
Only n=5 satisfies both statements.
SUFFICIENT.
The correct answer is
C.
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