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31 Jul 2025, 08:40
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Let \([x]\) represent the greatest integer less than or equal to \(x\). If \(n\) is a positive integer such that \([\frac{n}{4}] - [\frac{n}{9}] = 2\), what is the minimum possible value of \(n\)?
A. 7
B. 8
C. 9
D. 12
E. 19
A. 7
B. 8
C. 9
D. 12
E. 19
31 Jul 2025, 08:40
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Official Solution:
Let \([x]\) represent the greatest integer less than or equal to \(x\). If \(n\) is a positive integer such that \([\frac{n}{4}] - [\frac{n}{9}] = 2\), what is the minimum possible value of \(n\)?
A. 7
B. 8
C. 9
D. 12
E. 19
Notice that as n increases, both \([\frac{n}{4}]\) and \([\frac{n}{9}]\) increase, but \([\frac{n}{4}]\) increases faster. Every time \(n\) reaches a multiple of 4, \([\frac{n}{4}]\) increases by 1. Similarly, \([\frac{n}{9}]\) increases by 1 at multiples of 9. For example, for values \(n = 1\) to 3, both \(\frac{n}{4}\) and \(\frac{n}{9}\) are less than 1, so \([\frac{n}{4}]\) and \([\frac{n}{9}]\) are both 0. When \(n\) reaches 4, \([\frac{n}{4}]\) becomes 1 while \([\frac{n}{9}]\) remains 0, giving a difference of 1.
Now, for a difference of 2, we need a value of \(n\) for which \(\frac{n}{4}\) is 2 while \(\frac{n}{9}\) is still 0. Checking the next multiple of 4, which is 8, we get \([\frac{8}{4}] = 2\) and \([\frac{8}{9}] = 0\), giving the required difference of 2. So the minimum value of \(n\) is 8.
Answer: B
Let \([x]\) represent the greatest integer less than or equal to \(x\). If \(n\) is a positive integer such that \([\frac{n}{4}] - [\frac{n}{9}] = 2\), what is the minimum possible value of \(n\)?
A. 7
B. 8
C. 9
D. 12
E. 19
Notice that as n increases, both \([\frac{n}{4}]\) and \([\frac{n}{9}]\) increase, but \([\frac{n}{4}]\) increases faster. Every time \(n\) reaches a multiple of 4, \([\frac{n}{4}]\) increases by 1. Similarly, \([\frac{n}{9}]\) increases by 1 at multiples of 9. For example, for values \(n = 1\) to 3, both \(\frac{n}{4}\) and \(\frac{n}{9}\) are less than 1, so \([\frac{n}{4}]\) and \([\frac{n}{9}]\) are both 0. When \(n\) reaches 4, \([\frac{n}{4}]\) becomes 1 while \([\frac{n}{9}]\) remains 0, giving a difference of 1.
Now, for a difference of 2, we need a value of \(n\) for which \(\frac{n}{4}\) is 2 while \(\frac{n}{9}\) is still 0. Checking the next multiple of 4, which is 8, we get \([\frac{8}{4}] = 2\) and \([\frac{8}{9}] = 0\), giving the required difference of 2. So the minimum value of \(n\) is 8.
Answer: B
