There is a remainder of 4 when a number is divided by 7.what will be the remainder. if the square of the same number is divided by 7?
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Understanding Remainders When Squaring a Number
This problem involves understanding how remainders behave when we perform arithmetic operations like squaring a number. We are given information about the remainder when a number is divided by 7, and we need to find the remainder of the square of that same number when divided by 7.
Representing the Number
Let the number be denoted by '\(N\)'. The problem states that when '\(N\)' is divided by 7, the remainder is 4. We can express this using modular arithmetic or algebraic representation.
Algebraically, this means '\(N\)' can be written in the form: \(N = 7k + 4\) where '\(k\)' is an integer (0, 1, 2, ...).
In modular arithmetic terms, this is written as: \(N \equiv 4 \pmod{7}\)
Calculating the Square of the Number
Now, we need to find the square of the number '\(N\)', which is '\(N^2\)'.
Using the algebraic form: \(N^2 = (7k + 4)^2\)
Let's expand this expression using the formula \((a+b)^2 = a^2 + 2ab + b^2\): \(N^2 = (7k)^2 + 2(7k)(4) + 4^2\) \(N^2 = 49k^2 + 56k + 16\)
Finding the Remainder of the Square
Our goal is to find the remainder when '\(N^2\)' is divided by 7. Let's look at each term in the expanded expression '\(49k^2 + 56k + 16\)':
- First term: '\(49k^2\)' Since 49 is a multiple of 7 (\(49 = 7 \times 7\)), '\(49k^2\)' is definitely divisible by 7. The remainder is 0. \(49k^2 = 7 \times (7k^2)\) So, \(49k^2 \equiv 0 \pmod{7}\).
- Second term: '\(56k\)' Similarly, 56 is also a multiple of 7 (\(56 = 7 \times 8\)). So, '\(56k\)' is divisible by 7. The remainder is 0. \(56k = 7 \times (8k)\) So, \(56k \equiv 0 \pmod{7}\).
- Third term: '\(16\)' We need to find the remainder when 16 is divided by 7. \(16 = (7 \times 2) + 2\) The remainder is 2. So, \(16 \equiv 2 \pmod{7}\).
To find the remainder of the entire expression '\(N^2 = 49k^2 + 56k + 16\)' when divided by 7, we add the remainders of each term: Remainder of \(N^2\) = (Remainder of \(49k^2\)) + (Remainder of \(56k\)) + (Remainder of \(16\)) Remainder of \(N^2\) = \(0 + 0 + 2\) Remainder of \(N^2\) = \(2\)
Alternatively, using modular arithmetic directly: If \(N \equiv 4 \pmod{7}\), then squaring both sides gives: \(N^2 \equiv 4^2 \pmod{7}\) \(N^2 \equiv 16 \pmod{7}\) Since \(16 = 2 \times 7 + 2\), we have \(16 \equiv 2 \pmod{7}\). Therefore, \(N^2 \equiv 2 \pmod{7}\).
Conclusion
The remainder when the square of the number is divided by 7 is 2.
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select the correct answer using the code given below:
