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If a and b are two odd prime numbers, then a2 - b2 is ____(prime/composite/neither prime nor composite).
- a)Prime Number
- b)Composite Number
- c)None
- d)Neither prime nor composite
Correct answer is option 'B'. Can you explain this answer?
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If a and b are two odd prime numbers, then a2 - b2 is ____(prime/comp...
Composite numbers can be defined as the whole numbers that have more than two factors. Whole numbers that are not prime are composite numbers, because they are divisible by more than two numbers.
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If a and b are two odd prime numbers, then a2 - b2 is ____(prime/comp...
Given:
- a and b are two odd prime numbers
To find:
- Whether a^2 - b^2 is prime, composite, or neither prime nor composite
Solution:
We are given that a and b are odd prime numbers. Let's understand the properties of odd prime numbers and then analyze the expression a^2 - b^2.
Odd Prime Numbers:
- An odd number is any integer that is not divisible evenly by 2.
- A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.
- Examples of odd prime numbers: 3, 5, 7, 11, 13, etc.
Expression a^2 - b^2:
- The expression a^2 - b^2 can be simplified as (a + b)(a - b) using the difference of squares formula.
- Since a and b are odd prime numbers, both a + b and a - b will be even numbers.
- An even number is any integer that is divisible evenly by 2.
Analyzing (a + b)(a - b):
- We know that both a + b and a - b are even numbers.
- An even number can be expressed as the product of 2 and some other integer.
- So, we can write a + b as 2x and a - b as 2y, where x and y are integers.
- Substituting these values in (a + b)(a - b), we get (2x)(2y) = 4xy.
- Therefore, the expression a^2 - b^2 can be written as 4xy, which is divisible evenly by 4.
Conclusion:
- The expression a^2 - b^2 is divisible evenly by 4.
- Any number that is divisible evenly by 4 is not a prime number.
- Therefore, a^2 - b^2 is a composite number.
- Hence, the correct answer is option 'B' (Composite Number).
- a and b are two odd prime numbers
To find:
- Whether a^2 - b^2 is prime, composite, or neither prime nor composite
Solution:
We are given that a and b are odd prime numbers. Let's understand the properties of odd prime numbers and then analyze the expression a^2 - b^2.
Odd Prime Numbers:
- An odd number is any integer that is not divisible evenly by 2.
- A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.
- Examples of odd prime numbers: 3, 5, 7, 11, 13, etc.
Expression a^2 - b^2:
- The expression a^2 - b^2 can be simplified as (a + b)(a - b) using the difference of squares formula.
- Since a and b are odd prime numbers, both a + b and a - b will be even numbers.
- An even number is any integer that is divisible evenly by 2.
Analyzing (a + b)(a - b):
- We know that both a + b and a - b are even numbers.
- An even number can be expressed as the product of 2 and some other integer.
- So, we can write a + b as 2x and a - b as 2y, where x and y are integers.
- Substituting these values in (a + b)(a - b), we get (2x)(2y) = 4xy.
- Therefore, the expression a^2 - b^2 can be written as 4xy, which is divisible evenly by 4.
Conclusion:
- The expression a^2 - b^2 is divisible evenly by 4.
- Any number that is divisible evenly by 4 is not a prime number.
- Therefore, a^2 - b^2 is a composite number.
- Hence, the correct answer is option 'B' (Composite Number).
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If a and b are two odd prime numbers, then a2 - b2 is ____(prime/composite/neither prime nor composite).a)Prime Numberb)Composite Numberc)Noned)Neither prime nor compositeCorrect answer is option 'B'. Can you explain this answer?
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