# Even Odd | Algebra - Number Theory

**Even Odd | Algebra - Number Theory**

If x and y are both odd numbers, which of the following is an odd number?

- A.
x + y

- B.
x + y + 1

- C.
xy - 1

- D.
xy + 1

- E.
None of these

Answer: Option B

**Explanation** :

Sum of even number of odd numbers is always even, and

Sum of odd number of odd numbers is always odd.

If x and y are both odd numbers then x + y will be even so, x + y + 1 will be odd.

Hence, option (b).

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**Even Odd | Algebra - Number Theory**

If x is an even number and y is an odd number then which of the following is even.

A. (y – x) × (x – y)

B. (x – y) × (x + y)

C. (x × y) × (y + x)

- A.
Only A and B

- B.
Only C

- C.
Only A and C

- D.
Only B and C

- E.
All A, B and C

Answer: Option B

**Explanation** :

A. (y – x) × (x – y) = odd × odd = odd

B. (x – y) × (x + y) = odd × odd = odd

C. (x × y) × (y + x) = even × odd = even

Hence, option (b).

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**Even Odd | Algebra - Number Theory**

If x is an even number and y is an odd number then which of the following is even.

A. x^{y} + y^{y}

B. y^{x} − x^{x}

C. y × x^{y}

- A.
Only A

- B.
Only B

- C.
Only C

- D.
Only A and B

- E.
Only B and C

Answer: Option C

**Explanation** :

A. x^{y} + y^{y} = even + odd = odd

B. y^{x} – x^{x} = odd – even = odd

C. y × x^{y} = odd × even = even

Hence, option (c).

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**Even Odd | Algebra - Number Theory**

If x is an even number and y is an odd number then which of the following is even.

A. y × (x + y) – y^{y}

B. x × (x – y) – x^{y}

C. x^{y} × (y^{y} – y) – x^{x}

- A.
Only A

- B.
Only A and B

- C.
Only A and C

- D.
Only B and C

- E.
All A, B and C

Answer: Option E

**Explanation** :

A. y × (x + y) – y^{y} = odd × odd – odd = even

B. x × (x – y) – x^{y} = even × odd – even = even

C. x^{y} × (y^{y} – y) – x^{x} = even × even – even = even

Hence, option (e).

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**Even Odd | Algebra - Number Theory**

If x is an even number and y is an odd number then which of the following is even.

A. (x – y) + (x^{y} + y) – x^{y}

B. (x – y) – (x^{y} – x) + x^{y}

C. (x + y) × (x^{y} + y) × x^{y}

- A.
Only B

- B.
Only A and B

- C.
Only B and C

- D.
Only A and C

- E.
Only A

Answer: Option D

**Explanation** :

A. (x – y) + (x^{y} + y) – x^{y} = odd + odd – even = even

B. (x – y) – (x^{y} – x) + x^{y} = odd – even + even = odd

C. (x + y) × (x^{y} + y) × x^{y} = odd × odd × even = even

Hence, option (d).

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**Even Odd | Algebra - Number Theory**

If x is an even number and y is an odd number then which of the following is even.

A. x^{x} + x^{y} + (y – x)

B. y^{x} + x^{y} – (y + x)

C. x^{x} + y^{y} + (x – y)

- A.
Only A and B

- B.
Only A and C

- C.
Only B

- D.
Only C

Answer: Option A

**Explanation** :

A. x_{x} + x_{y} + (y – x) = even + even + odd = odd

B. y_{x} + x_{y} – (y + x) = odd + even – odd = even

C. x_{x} + y_{y} + (x – y) = even + odd + odd = even

Hence, option (a).

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**Even Odd | Algebra - Number Theory**

Given positive odd integers x, y and z, which of the following is not necessarily true?

- A.
x

^{2}y^{2}z^{2}is odd - B.
2(x

^{2}+ x^{3})z^{2}is even - C.
5x + y + z

^{4}is odd - D.
$\frac{{z}^{2}({x}^{4}+{y}^{4})}{2}$ is even

Answer: Option D

**Explanation** :

**Option 1**: is a product of three odd numbers and so is odd.

**Option 2**: x^{2} + x^{3} is even.

∴ The given product is even.

**Option 3**: Is a sum of three odd numbers and so is odd.

**Option 4**: If x, y and z are odd, then x^{4} + y^{4} will be even.

z^{2} is another odd number.

∴ Depending on whether x^{4} + y^{4} is a multiple of 4 or not,

$\frac{{z}^{2}({x}^{4}+{y}^{4})}{2}$ may be even or odd.

Hence, option (d).

Workspace:

**Even Odd | Algebra - Number Theory**

Let x, y and z be distinct integers, that are odd and positive. Which one of the following statements cannot be true?

- A.
xyz

^{2}is odd - B.
(x − y)

^{2}z is even - C.
(x + y − z)

^{2}(x + y) is even - D.
(x − y) (y + z) (x + y − z) is odd

Answer: Option D

**Explanation** :

**Option 1**: Product of odd numbers is always odd.

∴ x × y × z × z is always odd

**Option 2**: (x − y) is even.

∴ (x − y)^{2}z is even.

**Option 3**: (x + y) is always even.

∴ (x + y − z)^{2}(x + y) is even.

**Option 4**: (x − y) is even, (y + z) is even, and (x + y − z) is odd.

∴ (x − y) (y + z) (x + y − z) is even.

Hence, option (d).

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**Even Odd | Algebra - Number Theory**

Let x and y be positive integers such that x is prime and y is composite. Then,

- A.
(y – x) cannot be an even integer

- B.
xy cannot be an even integer

- C.
(x + y)/x cannot be an even integer

- D.
None of these

Answer: Option D

**Explanation** :

**Option 1**: False

∵ (y – x) is equal to an even integer when x = 2 and y = 6

**Option 2**: False

∵ xy is an even integer whenever x = 2 and y = 6

**Option 3**: False

∵ (x + y)/x is an even integer when x = 3 and y = 6, 9, 15, 21, 27… and so on. (i.e. Taking any value of x and then taking y as a multiple of that x will work.)

Hence, option (d).

Workspace:

**Even Odd | Algebra - Number Theory**

If a, b and c are positive integers, then (a + b)(b + c)(c + a) is even or odd?

Mark your answer as:

1, if even

2, if odd

3, if the answer can’t be determined

Answer: 1

**Explanation** :

**Case 1**: When all three of a, b and c are even the given product will be even.

**Case 2**: When all three of a, b and c are odd the given product will be even.

**Case 3**: When 1 of a, b and c is even and other 2 are odd the given product will be even.

If a is odd whereas b and c are even, (b + c) will be even making the product even.

**Case 4**: When 1 of a, b and c is odd and other 2 are even the given product will be even.

If a is even whereas b and c are odd, (b + c) will be even making the product even.

Hence, the given product is always even.

Hence, 1.

Workspace:

**Even Odd | Algebra - Number Theory**

The sum of 100 natural numbers will be:

Type in your answer as

1 for even

2 for odd

3 for cannot be determined

Answer: 3

**Explanation** :

If all 100 number are even, the sum will be even.

If 99 of these numbers are even and one is odd, the sum will be odd.

∴ We cannot be sure whether the sum will be even or odd.

Hence, 3.

Workspace:

**Even Odd | Algebra - Number Theory**

The sum of 100 odd natural numbers will be:

Type in your answer as

1 for even

2 for odd

3 for cannot be determined

Answer: 1

**Explanation** :

Sum of an odd number of odd numbers will always be odd.

Sum of an even number of odd numbers will always be even.

Hence, 2.

Workspace:

**Even Odd | Algebra - Number Theory**

The sum of 77 even natural numbers will be:

Type in your answer as

1 for even

2 for odd

3 for cannot be determined

Answer: 1

**Explanation** :

Sum of any number of even numbers will always be even.

Hence, 1.

Workspace:

**Even Odd | Algebra - Number Theory**

Two different prime numbers X and Y, both are greater than 10, then which of the following must be true?

- A.
X + Y > 30

- B.
X - Y ≠ 25

- C.
Either (a) or (b)

- D.
None of these

Answer: Option B

**Explanation** :

X and Y are two prime numbers greater than 10. Hence, both X and Y must be odd.

Now difference of two odd numbers will always be even. Hence option (b) is true.

Hence, option (b).

Workspace:

**Even Odd | Algebra - Number Theory**

x, y and z are distinct prime numbers in ascending orders such that x + y + z is even. Which of the following is true?

- A.
(x - y) × z is even

- B.
(y - z) × x is odd

- C.
(z - x) × y is even

- D.
(x + y) × z is odd

- E.
None of these

Answer: Option D

**Explanation** :

Since, sum of three distinct numbers is even, there are two possibilities

**Case 1**: All three numbers are even.

There is only one even prime number i.e., 2.

Hence, this case is not possible.

**Case 2**: One of the three numbers are even and remaining two are odd.

There is only one even prime number i.e., 2. Hence, x = 2 and y and z are two distinct prime numbers greater than 2.

Now, checking for options:

Option (d): (x + y) × z = (even + odd) × odd = odd × odd = odd

Hence, option (d).

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