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If a prime number on division by 4 gives a remainder of 1, then that number can be expressed as
- a)sum of squares of two natural numbers
- b)sum of cubes of two natural numbers
- c)sum of square roots of two natural numbers
- d)sum of cube roots of two natural numbers
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If a prime number on division by 4 gives a remainder of 1, then that n...
Explanation:
To prove that a prime number, when divided by 4, gives a remainder of 1, can be expressed as the sum of squares of two natural numbers, we need to follow the given steps.
Step 1: Prime Number
Let's consider a prime number, p, which when divided by 4 gives a remainder of 1. So, we can write it as p = 4k + 1, where k is an integer.
Step 2: Expressing p as the Sum of Squares
Our goal is to express p as the sum of squares of two natural numbers. Let's assume that p can be expressed as the sum of squares of two natural numbers, a^2 and b^2, where a and b are natural numbers.
We can write the equation as p = a^2 + b^2.
Step 3: Rearranging the Equation
Rearranging the above equation, we get p - a^2 = b^2.
Step 4: Considering Modulo 4
Now, let's consider the modulo 4 for both sides of the equation.
p - a^2 ≡ b^2 (mod 4)
Since p ≡ 1 (mod 4), we can substitute the value of p in the equation.
1 - a^2 ≡ b^2 (mod 4)
Step 5: Squares Modulo 4
Next, we need to consider the possible values of squares modulo 4.
Any perfect square can only be congruent to 0 or 1 modulo 4. This means that a^2 ≡ 0 or 1 (mod 4), and b^2 ≡ 0 or 1 (mod 4).
Step 6: Substituting Values
Substituting the possible values of a^2 and b^2 in the equation, we get the following possibilities:
1 - 0 ≡ 0 (mod 4) - This is not possible as it contradicts our assumption that p is a prime number.
1 - 0 ≡ 1 (mod 4) - This is not possible as it contradicts our assumption that p is a prime number.
1 - 1 ≡ 0 (mod 4) - This is not possible as it contradicts our assumption that p is a prime number.
1 - 1 ≡ 1 (mod 4) - This satisfies our assumption that p is a prime number.
Conclusion:
From the above analysis, we can conclude that a prime number, when divided by 4 and gives a remainder of 1, can be expressed as the sum of squares of two natural numbers.
Therefore, the correct answer is option 'A' - sum of squares of two natural numbers.
To prove that a prime number, when divided by 4, gives a remainder of 1, can be expressed as the sum of squares of two natural numbers, we need to follow the given steps.
Step 1: Prime Number
Let's consider a prime number, p, which when divided by 4 gives a remainder of 1. So, we can write it as p = 4k + 1, where k is an integer.
Step 2: Expressing p as the Sum of Squares
Our goal is to express p as the sum of squares of two natural numbers. Let's assume that p can be expressed as the sum of squares of two natural numbers, a^2 and b^2, where a and b are natural numbers.
We can write the equation as p = a^2 + b^2.
Step 3: Rearranging the Equation
Rearranging the above equation, we get p - a^2 = b^2.
Step 4: Considering Modulo 4
Now, let's consider the modulo 4 for both sides of the equation.
p - a^2 ≡ b^2 (mod 4)
Since p ≡ 1 (mod 4), we can substitute the value of p in the equation.
1 - a^2 ≡ b^2 (mod 4)
Step 5: Squares Modulo 4
Next, we need to consider the possible values of squares modulo 4.
Any perfect square can only be congruent to 0 or 1 modulo 4. This means that a^2 ≡ 0 or 1 (mod 4), and b^2 ≡ 0 or 1 (mod 4).
Step 6: Substituting Values
Substituting the possible values of a^2 and b^2 in the equation, we get the following possibilities:
1 - 0 ≡ 0 (mod 4) - This is not possible as it contradicts our assumption that p is a prime number.
1 - 0 ≡ 1 (mod 4) - This is not possible as it contradicts our assumption that p is a prime number.
1 - 1 ≡ 0 (mod 4) - This is not possible as it contradicts our assumption that p is a prime number.
1 - 1 ≡ 1 (mod 4) - This satisfies our assumption that p is a prime number.
Conclusion:
From the above analysis, we can conclude that a prime number, when divided by 4 and gives a remainder of 1, can be expressed as the sum of squares of two natural numbers.
Therefore, the correct answer is option 'A' - sum of squares of two natural numbers.
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If a prime number on division by 4 gives a remainder of 1, then that number can be expressed asa)sum of squares of two natural numbersb)sum of cubes of two natural numbersc)sum of square roots of two natural numbersd)sum of cube roots of two natural numbersCorrect answer is option 'A'. Can you explain this answer?
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