petrified17 wrote:
If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by which of following(s)?
I. 3
II. 6
III. 8
A. I
B. I and II
C. II and III
D. II
E. III
Roman Numeral I:
If m is 1, then k = 1 x 5 x 6.
If m is 2, k = 2 x 6 x 7.
If m is 3, k = 3 x 7 x 9.
In all three cases, we see that m*(m+4)*(m+5) must be divisible by 3. A more dynamic proof to show that k is divisible by 3 is to let m = 3q + r where q is an integer and r = 0, 1 or 2.
If m = 3q, obviously k is divisible by 3 since m is a multiple of 3.
If m = 3q + 1, then k is divisible by 3 since m + 5 = 3q + 6 is a multiple of 3.
If m = 3q + 2, then k is divisible by 3 since m + 4 = 3q + 6 is a multiple of 3.
Hence, k is always divisible by 3, regardless of the value of integer m.
Roman Numeral II:
We observe that m + 4 and m + 5 are two consecutive integers, hence, one of them is necessarily even. We know that k is even and from Roman Numeral I, we also know that k is divisible by 3. Thus, k must be divisible by 6.
Roman Numeral III:
We notice that m and m + 4 have the same parity (they are both odd or both even) and the parity of m + 5 is opposite to that of m (if m is even, m + 5 is odd and if m is odd, m + 5 is even). When we have a product of three integers, the product can be divisible by 8 if a) one of the integers is divisible by 8, b) one of the integers is divisible by 4 and one of the integers is divisible by 2 or c) each of the integers is divisible by 2.Since the parity of m + 5 is opposite to the remaining integers, it is impossible that each of m, m + 4 and m + 5 are divisible by 2. So if we pick a value for m such that neither of m, m + 4 and m + 5 are divisible by 4, the product cannot be divisible by 8. Such a value is m = 1, in which case k = 1*5*6 = 30. As we can see, k is not necessarily divisible by 8.
Answer: B
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