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16 Sep 2014, 01:11
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If \(a\), \(b\), and \(c\) are consecutive integers, which of the following could be true?
I. \(abc\) is divisible by 6
II. \(a + c - b\) is odd
III. \(\frac{c}{a} \gt b\)
A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
I. \(abc\) is divisible by 6
II. \(a + c - b\) is odd
III. \(\frac{c}{a} \gt b\)
A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
16 Sep 2014, 01:11
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Official Solution:
If \(a\), \(b\), and \(c\) are consecutive integers, which of the following could be true?
I. \(abc\) is divisible by 6
II. \(a + c - b\) is odd
III. \(\frac{c}{a} \gt b\)
A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
Firstly, it's important to note that \(a\), \(b\), and \(c\) being consecutive integers does not imply that \(a < b < c\). They could be in any order. Secondly, we're asked to find which options COULD be true, not which MUST be true.
I. \(abc\) is divisible by 6.
This statement is always true as, in any set of three consecutive integers, at least one number will certainly be even, and one will certainly be a multiple of 3. Therefore, their product will definitely be divisible by \(2*3 = 6\).
P.S. In general, the product of \(n\) consecutive integers is always divisible by \(n!\). For instance, the product of any four consecutive integers will always be divisible by \(4!=24\).
II. \(a + c - b\) is odd.
This statement could be true if the values for \(a\), \(b\), and \(c\) form a sequence like {even, odd, even}, irrespective of which variable represents which value. For example, if \(a = 2\), \(b=3\), and \(c=4\), then \(a + c - b=3\), which is odd.
III. \(\frac{c}{a} \gt b\).
This statement could also be true. For instance, consider \(a = 1\), \(b=2\), and \(c=3\).
Answer: E
If \(a\), \(b\), and \(c\) are consecutive integers, which of the following could be true?
I. \(abc\) is divisible by 6
II. \(a + c - b\) is odd
III. \(\frac{c}{a} \gt b\)
A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
Firstly, it's important to note that \(a\), \(b\), and \(c\) being consecutive integers does not imply that \(a < b < c\). They could be in any order. Secondly, we're asked to find which options COULD be true, not which MUST be true.
I. \(abc\) is divisible by 6.
This statement is always true as, in any set of three consecutive integers, at least one number will certainly be even, and one will certainly be a multiple of 3. Therefore, their product will definitely be divisible by \(2*3 = 6\).
P.S. In general, the product of \(n\) consecutive integers is always divisible by \(n!\). For instance, the product of any four consecutive integers will always be divisible by \(4!=24\).
II. \(a + c - b\) is odd.
This statement could be true if the values for \(a\), \(b\), and \(c\) form a sequence like {even, odd, even}, irrespective of which variable represents which value. For example, if \(a = 2\), \(b=3\), and \(c=4\), then \(a + c - b=3\), which is odd.
III. \(\frac{c}{a} \gt b\).
This statement could also be true. For instance, consider \(a = 1\), \(b=2\), and \(c=3\).
Answer: E
