Question

Let \(N\), \(x\) and \(y\) be positive integers such that \(N=x+y\), \(2<x<10\) and \(14<y<23\). If \(N>25\), then how many distinct values are possible for \(N\)? [This Question was asked as TITA]

• A. 4
• B. 7
• C. 5
• D. 6

Answer 1

The Correct Option is D

We are given the conditions: \[ N = x + y,\quad 2 < x < 10,\quad 14 < y < 23,\quad N > 25 \]

Step 1: Identify possible values for \(x\) and \(y\)
Valid integers for \(x\): \(x = 3, 4, 5, 6, 7, 8, 9\)
Valid integers for \(y\): \(y = 15, 16, 17, 18, 19, 20, 21, 22\)

Step 2: Calculate combinations where \(N = x + y > 25\)
We compute values of \(N\) only when the sum is strictly greater than 25.

• \(x = 3\): \(y = 23 \Rightarrow N = 26\)
• \(x = 4\): \(y = 22, 23 \Rightarrow N = 26, 27\)
• \(x = 5\): \(y = 21, 22, 23 \Rightarrow N = 26, 27, 28\)
• \(x = 6\): \(y = 20, 21, 22, 23 \Rightarrow N = 26, 27, 28, 29\)
• \(x = 7\): \(y = 19, 20, 21, 22, 23 \Rightarrow N = 26, 27, 28, 29, 30\)
• \(x = 8\): \(y = 18, 19, 20, 21, 22, 23 \Rightarrow N = 26, 27, 28, 29, 30, 31\)
• \(x = 9\): \(y = 17, 18, 19, 20, 21, 22, 23 \Rightarrow N = 26, 27, 28, 29, 30, 31, 32\)

Step 3: Collect distinct values of \(N\)
From all combinations above, the distinct values of \(N\) are:
\[ \{26, 27, 28, 29, 30, 31, 32\} \] Total distinct values = 7

Final Correction (if needed):
On reviewing constraints again: \(y < 23\), so \(y = 23\) is invalid.
Thus, we eliminate all combinations where \(y = 23\). Let's recalculate:

• \(x = 3\): No valid \(y\)
• \(x = 4\): \(y = 22 \Rightarrow N = 26\)
• \(x = 5\): \(y = 21, 22 \Rightarrow N = 26, 27\)
• \(x = 6\): \(y = 20, 21, 22 \Rightarrow N = 26, 27, 28\)
• \(x = 7\): \(y = 19, 20, 21, 22 \Rightarrow N = 26, 27, 28, 29\)
• \(x = 8\): \(y = 18, 19, 20, 21, 22 \Rightarrow N = 26, 27, 28, 29, 30\)
• \(x = 9\): \(y = 17, 18, 19, 20, 21, 22 \Rightarrow N = 26, 27, 28, 29, 30, 31\)

Distinct values of \(N\): \[ \{26, 27, 28, 29, 30, 31\} \] Total = 6 distinct values.

Answer: The number of distinct possible values of \(N\) is: \[ \boxed{6} \]

Answer 2

We are given the following constraints:

• \(2 < x < 10 \Rightarrow x \in \{3, 4, 5, 6, 7, 8, 9\}\)
• \(14 < y < 23 \Rightarrow y \in \{15, 16, 17, 18, 19, 20, 21, 22\}\)
• \(N = x + y\), with the additional condition that \(N > 25\)

Let's find the maximum possible value of \(N\):

The maximum \(x = 9\) and maximum \(y = 22\) give: 
\[ N_{\text{max}} = x + y = 9 + 22 = 31 \]

Now we compute values of \(N = x + y\) starting from maximum:

• \(x = 9,\ y = 22 \Rightarrow N = 31\)
• \(x = 9,\ y = 21 \Rightarrow N = 30\)
• \(x = 9,\ y = 20 \Rightarrow N = 29\)
• \(x = 9,\ y = 19 \Rightarrow N = 28\)
• \(x = 9,\ y = 18 \Rightarrow N = 27\)
• \(x = 9,\ y = 17 \Rightarrow N = 26\)
• \(x = 9,\ y = 16 \Rightarrow N = 25\)

However, we are told \(N > 25\), so we exclude \(N = 25\).

Therefore, the valid distinct values of \(N\) are: 
\[ \{26, 27, 28, 29, 30, 31\} \] So, there are a total of \(\boxed{6}\) distinct possible values for \(N\) that satisfy all conditions.