If a, b, and k are positive integers, is the sum (a + b) an even number or an odd number?(1) a = ( k3 + 3k2 + 3k + 6) (2) b = (k2 + 4a +5)a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.d)EACH statement ALONE is sufficient.e)Statements (1) and (2) TOGETHER are NOT sufficient.Correct answer is option 'C'. Can you explain this answer?

Question

Answer

To determine whether the sum (a + b) is even or odd, we need to consider the parity of each individual term (a and b). Let's analyze each statement separately:

Statement (1): a = k^3 + 3k^2 + 3k + 6

- From this equation, we can see that a is expressed as a sum of four terms: k^3, 3k^2, 3k, and 6.
- The first term, k^3, is always odd for any positive integer k.
- The second term, 3k^2, is always divisible by 3, so it is always even.
- The third term, 3k, is always divisible by 3, so it is always even.
- The fourth term, 6, is even since it is divisible by 2.
- Therefore, the sum of these four terms, a, will always be even.
- Statement (1) alone is sufficient to determine that (a + b) is always even.

Statement (2): b = k^2 + 4a - 5

- From this equation, we can see that b is expressed as a sum of three terms: k^2, 4a, and -5.
- The first term, k^2, can be either even or odd depending on the value of k.
- The second term, 4a, is always divisible by 4, so it is always even.
- The third term, -5, is odd since it is not divisible by 2.
- Therefore, the sum of these three terms, b, can be either even or odd depending on the value of k.
- Statement (2) alone is not sufficient to determine the parity of (a + b).

Considering both statements together:

- From statement (1), we know that a is always even.
- From statement (2), we know that b can be either even or odd depending on the value of k.
- Since a is always even, and the sum of two even numbers is always even, we can conclude that (a + b) will always be even.
- Both statements together are sufficient to determine that (a + b) is always even.

Therefore, the correct answer is option C - BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient to determine the parity of (a + b).

Answer

Steps 1 & 2: Understand Question and Draw Inferences

We are given that a, b, and k are positive integers. And, we have to find out whether the sum of a and b will be even or odd.

Let us draw a table for all the possible cases for the even-odd property of a and b, and the corresponding property of the sum (a+b):

Thus we see that:
If a and b are both odd or both even, then the sum (a+b) will be even

But if one out of a and b is even, and the other is odd, the sum (a+b) will be odd.

So, we will need to determine the even-odd property of both a and b in order to be in a position to say whether the sum (a+b) is odd or even.

Step 3: Analyze Statement 1

Let’s analyze statement I:

a = k³ + 3k² + 3k + 6

As we can see the value of a depends on the value of k. There can be two cases for the value of k – k can be either odd or even:

1. If k is odd:

This means, k³ = Odd (Odd*Odd*Odd = Odd; Think: 3*3*3 = 27)

Similarly, k² will also be odd

So, 3k² = Odd (Since 3 is odd and Odd*Odd = Odd)

Also, 3k = Odd

So, we get that

a = Odd number + odd number + odd number + even number

Think of easier numbers of the type of the right hand side of the above equation:

1 + 1 + 1 + 2 = 5, which is odd

This means that the RHS of the above equation will be an odd number

So, a is an odd number

So, if k is an odd number, a will be an odd number.

2. If k is even:

This means, k³ = Even (Even*Even*Even = Even; Think: 2*2*2 = 8)

Similarly, k² will also be even

So, 3k² = Even (Odd*Even = Even)

Also, 3k = Even

So, we get that

a = Even number + even number + even number + even number = Even number

So, a is an even number

Thus, if k is an even number, “a” will be an even number.

To conclude,

If k is odd => a is odd

If k is even => a is even

Thus, the value of a can be even or odd, depending on the value of k. This step fails to give us a definite answer whether a is an even number or odd number. Also, it does not give us any information about the values of b.

So, we can’t determine whether the expression a + b will be even or odd.

So, Statement 1 alone is not sufficient to arrive at a unique answer.

Step 4: Analyze Statement 2

Moving on, let’s analyze statement II:

b = (k² + 4a +5) …………… (2)

Now, the value of b depends on the values of a and k.

If the values of a and k are independent of each other, then, we will have to consider 4 cases for the value of b:

Thus, we see that:

i) When k is even, b is odd

ii) When k is odd, b is even

iii) The even-odd property of b doesn’t depend on the value of a

Now, it is possible that the values of k and a are dependent on one another.

For example, if you were given that k = 8a

This would mean that irrespective of whether a is even or odd, k will always be even.

In that case, you would have been able to determine for sure that b will always be odd (though you would still be clueless about whether a is odd or even).

Or, if you were given that a = 8k, you would have been able to determine that a is even no matter what the value of k.

We are not given any information about the relationship, if any, between k and a. Neither are we given any other clue about the even-odd property of a.

Since this statement doesn’t tell us whether:

i) k is even or odd

ii) a is even or odd

iii) a and k are independent or not,

we will not be able to determine whether the sum (a+b) is even or odd.

So, Statement 2 alone is not sufficient to arrive at a unique answer

Step 5: Analyze Both Statements Together (if needed)

Since the individual analysis of the statements does not yield us any significant result, let’s analyse both of them together:

From Statement 1:

If k is odd => a is odd

If k is even => a is even

From Statement 2:

When k is even, b is odd

When k is odd, b is even

Combining the two statements, we get the following table:

Thus, we see that irrespective of the value of k, the sum (a+b) is always odd.

Thus, we have been able to arrive at a unique solution by combining the two statements.

Answer: Option (C)

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