To determine if a two-digit number is odd, we need to analyze the information provided in Statement 1 (S1) and Statement 2 (S2).

Two-Digit Number Definition

Let the two-digit number be represented as \(N\). A two-digit number ranges from 10 to 99. We can express \(N\) as \(10a + b\), where \(a\) is the tens digit (\(a \in \{1, 2, ..., 9\}\)) and \(b\) is the units digit (\(b \in \{0, 1, ..., 9\}\)).

A number is considered odd if its units digit (\(b\)) is one of 1, 3, 5, 7, or 9.

A number is considered even if its units digit (\(b\)) is one of 0, 2, 4, 6, or 8.

Analyzing Statement 1 (S1): Sum of the Digits is Even

Statement 1 provides the condition that the sum of the digits, \(a + b\), is an even number.

For the sum of two integers to be an even number, there are exactly two possible scenarios for the parity (odd or even) of the individual digits:

• Case 1: Both digits \(a\) and \(b\) are even.
  • If \(a\) is an even digit (e.g., 2, 4, 6, 8) and \(b\) is an even digit (e.g., 0, 2, 4, 6, 8), then their sum \(a+b\) will be an even number.
  • In this situation, the two-digit number \(N = 10a + b\). Since \(10a\) will always be an even number (as 10 is even) and \(b\) is specified as even, their sum \(N\) will be an even number.
  • Example: Consider the two-digit number \(N = 20\). Here, \(a=2\) (even) and \(b=0\) (even). The sum of digits is \(2+0=2\), which is even. The number 20 is an even number.
  • Example: Consider the two-digit number \(N = 42\). Here, \(a=4\) (even) and \(b=2\) (even). The sum of digits is \(4+2=6\), which is even. The number 42 is an even number.
• Case 2: Both digits \(a\) and \(b\) are odd.
  • If \(a\) is an odd digit (e.g., 1, 3, 5, 7, 9) and \(b\) is an odd digit (e.g., 1, 3, 5, 7, 9), then their sum \(a+b\) will also be an even number.
  • In this situation, the two-digit number \(N = 10a + b\). Since \(10a\) will always be an even number and \(b\) is specified as odd, their sum \(N\) will be an odd number.
  • Example: Consider the two-digit number \(N = 11\). Here, \(a=1\) (odd) and \(b=1\) (odd). The sum of digits is \(1+1=2\), which is even. The number 11 is an odd number.
  • Example: Consider the two-digit number \(N = 35\). Here, \(a=3\) (odd) and \(b=5\) (odd). The sum of digits is \(3+5=8\), which is even. The number 35 is an odd number.

Since Statement 1 allows for the two-digit number \(N\) to be both odd (e.g., 11) and even (e.g., 20) while satisfying the condition, Statement 1 alone is not sufficient to definitively answer whether the two-digit number is odd.

Analyzing Statement 2 (S2): The Number, When Multiplied by 2, Fetches an Even Number

Statement 2 states that \(2 \times N\) is an even number.

• The property of multiplication dictates that if any integer \(N\) is multiplied by 2, the result will always be an even number, regardless of whether \(N\) itself is odd or even.
• Example: If \(N = 7\) (an odd number), then \(2 \times 7 = 14\), which is an even number.
• Example: If \(N = 8\) (an even number), then \(2 \times 8 = 16\), which is an even number.

This statement provides no unique information that helps us determine the parity (odd or even) of \(N\) itself. It is a universal truth for all integers.

Therefore, Statement 2 alone is not sufficient to answer whether the two-digit number is odd.

Combining Both Statements (S1 & S2)

When we combine Statement 1 and Statement 2, we consider the implications of both conditions simultaneously to see if a definitive answer can be reached.

As previously established, Statement 2 does not offer any useful additional information to determine if the two-digit number is odd, because multiplying any integer by 2 always yields an even number.

Consequently, combining the statements essentially means we are only relying on the information provided by Statement 1.

From our analysis of Statement 1, we observed that:

• It allows for cases where the two-digit number \(N\) is even (e.g., \(N=20\), where \(a=2, b=0\), sum of digits \(2+0=2\) is even, and 20 is even).
• It also allows for cases where the two-digit number \(N\) is odd (e.g., \(N=11\), where \(a=1, b=1\), sum of digits \(1+1=2\) is even, and 11 is odd).

Since we can still find examples where the number is odd and examples where the number is even, even after considering both statements, we cannot definitively answer the question "Is the two-digit number odd?" with a simple "Yes" or "No".

Therefore, both Statement 1 and Statement 2 combined are not sufficient to answer the question.

Final Answer Conclusion

Based on the detailed analysis, neither Statement 1 alone, nor Statement 2 alone, nor both statements combined provide sufficient information to determine if the two-digit number in question is odd. We cannot give a definitive "Yes" or "No" answer.