Is the two digit number odd? Statement 1 (S1) : Sum of the digits is even Statement 2 (S2) : The number, when multiplied by 2, fetches an even number.
Both S1 & S2 combined are not sufficient to answer the question.
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.
| Statement | Condition | Example (Number N) | Is N odd? | Sufficiency |
|---|---|---|---|---|
| S1: Sum of digits (\(a+b\)) is even | \(a\) even, \(b\) even | \(N=20\) (\(2+0=2\)) | No (20 is even) | Not Sufficient (allows both odd and even numbers) |
| \(a\) odd, \(b\) odd | \(N=11\) (\(1+1=2\)) | Yes (11 is odd) | ||
| S2: The number, when multiplied by 2, fetches an even number (\(2N\) is even) | Always true for any integer N | \(N=10 \Rightarrow 20\) (even) \(N=11 \Rightarrow 22\) (even) |
Cannot determine N's parity | Not Sufficient (provides no information about N itself) |
| S1 & S2 Combined | Effectively same as S1 (S2 adds no value) | Can be \(11\) (odd) or \(20\) (even) | Cannot determine definitively | Not Sufficient |
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.
Important Questions from Data Sufficiency
Which of the following statement is correct?
I. The value of 1002 - 992 + 982 - 972 + 962 - 952 + 942 - 932 + ...... + 222 - 212 is 4840.
II. The value of
\(\left(k^2+ \frac{1}{k^2} \right) \left(k- \frac{1}{k} \right) \left(k^4+ \frac{1}{k^4} \right) \) \(\left(k+ \frac{1}{k} \right)\left(k^4- \frac{1}{k^4} \right)\) is \(k^{16}- \frac{1}{k^{16}}\).
Given below is a question followed by two statements I and II. Read both statements carefully to decide which one of them is sufficient to answer the question.
Six people - A, B, C, D, E and F are sitting in a straight line facing north. Who sits to the immediate left of D?
(I) A sits second from one of the extreme ends of the line. Only two people sit between A and B. E sits to the immediate right of C.
(II) C sits third from the left end of the line. Only one person sits between C and F. E sits third to the right of F.
Please read the below equation and select an appropriate option from the following.
x + y = 10 ; y = x - 2
Quantity A is x.
Quantity B is y.
A lady wants to buy the following items in the given price range-
1. Tomato between Rs. 40 and Rs. 45 per kg.
2. Grapes in the price range of Rs. 80 and Rs. 90 per kg.
3. Milk packets at Rs. 23 per liter.
In which shop from the following will she definitely get all her items/
A. Shop S sells tomato at Rs. 22.5 per half kg, Grapes at Rs. 82 per kg and milk at Rs. 24 per liter.
B. Shop H sells grapes at Rs. 21 per quarter kg, milk at 12.5 per half litre and tomato at Rs. 22 per half kg.
C. Shop O sells milk at Rs. 11.5 per half litre, tomato at 21 per half kg and grapes at Rs. 43 per half kg.
D. Shop P sells tomato at Rs. 23.5 per half kg, grapes at Rs. 85 per kg and milk at Rs. 23 per litre.You are given a question and two statements. Identify which of the statements is/are sufficient to answer the question.
Question:
When is Yuvi's wedding anniversary?
Statements:
1: On the 7 th day of a month.
2. The month has 29 days only in a leap year.
