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On dividing a number by 5, we get 3 as remainder. What will the remainder when the square of the this number is divided by 5 ?
- a)0
- b)1
- c)2
- d)4
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
On dividing a number by 5, we get 3 as remainder. What will the remain...
Let the number be x and on dividing x by 5, we get k as quotient and 3 as remainder.
∴ x = 5k + 3
∴ x2 = (5k +3)2
= (25k2 + 30k + 9)
= 5(5k2 + 6k+1) + 4
∴ on dividing x2 by 5, we get 4 as remainder.
∴ x = 5k + 3
∴ x2 = (5k +3)2
= (25k2 + 30k + 9)
= 5(5k2 + 6k+1) + 4
∴ on dividing x2 by 5, we get 4 as remainder.
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Community Answer
On dividing a number by 5, we get 3 as remainder. What will the remain...
Given Information:
- When a number is divided by 5, the remainder is 3.
To Find:
- The remainder when the square of this number is divided by 5.
Explanation:
Let's assume the number as 'x'.
When 'x' is divided by 5, the remainder is 3. This can be written as:
x ≡ 3 (mod 5)
This means 'x' leaves a remainder of 3 when divided by 5.
We need to find the remainder when the square of 'x' is divided by 5. This can be written as:
x^2 ≡ ? (mod 5)
Method 1: Using Algebraic Approach
To find the remainder of x^2 when divided by 5, we can square both sides of the congruence:
(x ≡ 3 (mod 5))^2
x^2 ≡ 3^2 (mod 5)
x^2 ≡ 9 (mod 5)
x^2 ≡ 4 (mod 5)
Therefore, the remainder when the square of the given number is divided by 5 is 4.
Method 2: Using Example
Let's take a specific example to understand it better.
Assume the number 'x' as 8. When 8 is divided by 5, the remainder is 3.
8 ÷ 5 = 1 remainder 3
Now, let's calculate the square of 8 and divide it by 5:
8^2 = 64
64 ÷ 5 = 12 remainder 4
Similarly, if we take any other number that leaves a remainder of 3 when divided by 5 and square it, we will always get a remainder of 4 when divided by 5.
Conclusion:
The remainder when the square of a number that leaves a remainder of 3 when divided by 5 is divided by 5 is always 4. Therefore, the correct answer is option 'D'.
- When a number is divided by 5, the remainder is 3.
To Find:
- The remainder when the square of this number is divided by 5.
Explanation:
Let's assume the number as 'x'.
When 'x' is divided by 5, the remainder is 3. This can be written as:
x ≡ 3 (mod 5)
This means 'x' leaves a remainder of 3 when divided by 5.
We need to find the remainder when the square of 'x' is divided by 5. This can be written as:
x^2 ≡ ? (mod 5)
Method 1: Using Algebraic Approach
To find the remainder of x^2 when divided by 5, we can square both sides of the congruence:
(x ≡ 3 (mod 5))^2
x^2 ≡ 3^2 (mod 5)
x^2 ≡ 9 (mod 5)
x^2 ≡ 4 (mod 5)
Therefore, the remainder when the square of the given number is divided by 5 is 4.
Method 2: Using Example
Let's take a specific example to understand it better.
Assume the number 'x' as 8. When 8 is divided by 5, the remainder is 3.
8 ÷ 5 = 1 remainder 3
Now, let's calculate the square of 8 and divide it by 5:
8^2 = 64
64 ÷ 5 = 12 remainder 4
Similarly, if we take any other number that leaves a remainder of 3 when divided by 5 and square it, we will always get a remainder of 4 when divided by 5.
Conclusion:
The remainder when the square of a number that leaves a remainder of 3 when divided by 5 is divided by 5 is always 4. Therefore, the correct answer is option 'D'.
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