[email protected] 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
Answer-C
Target question: What is the value of positive integer n?
Statement 1: For every positive integer m, the product m(m + 1)(m + 2) ... (m + n) is divisible by 16
First notice that m, m+1, m+2, m+3 etc are CONSECUTIVE INTEGERS
There's a nice rule says:
The product of k consecutive integers is divisible by k, k-1, k-2,...,2, and 1
So, for example, the product of any 5 consecutive integers will be divisible by 5, 4, 3, 2 and 1
NOTE: the product may be divisible by other numbers as well, but these divisors are guaranteed.
So, if the product of m, m+1, m+2 ... m+n is divisible by 16, there are many possible values of n. Consider these two conflicting cases:
Case a:
n = 15. This means that (m)(m+1)(m+2)...(m+n) is the product of 16 consecutive integers. So, by the
above rule, the product is definitely divisible by 16
Case b:
n = 16. This means that (m)(m+1)(m+2)...(m+n) is the product of 17 consecutive integers. So, by the
above rule, the product is definitely divisible by 16
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: n² - 9n + 20 = 0
Factor to get: (n - 4)(n - 5) = 0
So,
n = 4 or
n = 5
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 2 says that n = 4 or n = 5
However, although n = 4 satisfies statement 2, it does not necessarily satisfy statement 1.
If n = 4, then there are 5 consecutive integers in the product (m)(m+1)(m+2)...(m+n), and having 5 consecutive integers does not necessarily ensure that the product is divisible by 16.
For example (1)(2)(3)(4)(5) = 120, 120 is NOT divisible by 16.
So, if n ≠4, then
n MUST EQUAL 5 [since statement 2 tells us that n equals EITHER 4 OR 5]
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer =
C
Cheers,
Brent
