Even Odd | Algebra - Number Theory
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1. 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
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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).
2. 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)
(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
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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).
3. 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. xy + yy x − xx y
(a) Only A
(b) Only B
(c) Only C
(d) Only A and B
(e) Only B and C
Answer: Option
C
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Explanation :
A. xy + yy = even + odd = odd
B. yx – xx = odd – even = odd
C. y × xy = odd × even = even
Hence, option (c).
4. 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) – yy y y × (yy – y) – xx
(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
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Explanation :
A. y × (x + y) – yy = odd × odd – odd = even
B. x × (x – y) – xy = even × odd – even = even
C. xy × (yy – y) – xx = even × even – even = even
Hence, option (e).
5. 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) + (xy + y) – xy y – x) + xy y + y) × xy
(a) Only B
(b) Only A and B
(c) Only B and C
(d) Only A and C
(e) Only A
Answer: Option
D
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Explanation :
A. (x – y) + (xy + y) – xy = odd + odd – even = even
B. (x – y) – (xy – x) + xy = odd – even + even = odd
C. (x + y) × (xy + y) × xy = odd × odd × even = even
Hence, option (d).
6. 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. xx + xy + (y – x)x + xy – (y + x)x + yy + (x – y)
(a) Only A and B
(b) Only A and C
(c) Only B and C
(d) Only C
Answer: Option
C
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Explanation :
A. xx + xy + (y – x) = even + even + odd = odd
B. yx + xy – (y + x) = odd + even – odd = even
C. xx + yy + (x – y) = even + odd + odd = even
Hence, option (c).
7. Even Odd | Algebra - Number Theory
Given positive odd integers x, y and z, which of the following is not necessarily true?
(a) x2 y2 z2 is odd
(b) 2(x2 + x3 )z2 is even
(c) 5x + y + z4 is odd
(d) z 2 ( x 4 + y 4 ) 2
Answer: Option
D
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Explanation :
Option 1 : is a product of three odd numbers and so is odd.
Option 2 : x2 + x3 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 x4 + y4 will be even.
z2 is another odd number.
∴ Depending on whether x4 + y4 is a multiple of 4 or not,
z 2 ( x 4 + y 4 ) 2
Hence, option (d).
8. 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?
Answer: Option
D
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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).
9. 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
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Explanation :
Option 1 : False
Option 2 : False
Option 3 : False
Hence, option (d).
10. 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:
Answer: 1
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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.
Case 4 : When 1 of a, b and c is odd and other 2 are even the given product will be even.
Hence, the given product is always even.
Hence, 1.
11. Even Odd | Algebra - Number Theory
The sum of 100 natural numbers will be:
Type in your answer as
Answer: 3
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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.
12. Even Odd | Algebra - Number Theory
The sum of 100 odd natural numbers will be:
Type in your answer as
Answer: 1
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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, 1.
13. Even Odd | Algebra - Number Theory
The sum of 77 even natural numbers will be:
Type in your answer as
Answer: 1
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Explanation :
Sum of any number of even numbers will always be even.
Hence, 1.
14. 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
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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).
15. 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
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Explanation :
Since, sum of three distinct numbers is even, there are two possibilities
Case 1 : All three numbers are even.
Case 2 : One of the three numbers are even and remaining two are odd.
Now, checking for options:
Option (d): (x + y) × z = (even + odd) × odd = odd × odd = odd
Hence, option (d).
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