A number n is divided by 21gives remainder 5find the remainder when it...
A number n is divided by 21gives remainder 5find the remainder when it...
Remainder when a number is divided by 7
To find the remainder when a number is divided by 7, we can use the concept of modular arithmetic.
Modular arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value called the modulus. In our case, the modulus is 7.
For any integer n, we can represent it as:
n = q * m + r
Where:
- n is the dividend
- q is the quotient
- m is the divisor (modulus)
- r is the remainder
Given Information
We are given that a number n, when divided by 21, gives a remainder of 5.
n = 21q + 5
Using modular arithmetic
We can rewrite the equation using modular arithmetic:
n ≡ 5 (mod 21)
This means that n is congruent to 5 modulo 21.
To find the remainder when n is divided by 7, we need to determine the congruence of n modulo 7.
Applying the properties of congruence
We know that if two numbers are congruent modulo m, their difference is also congruent modulo m.
n ≡ 5 (mod 21)
Subtracting 5 from both sides:
n - 5 ≡ 0 (mod 21)
Now, we need to find an equivalent congruence modulo 7.
Using the properties of modular arithmetic
We can rewrite the congruence modulo 7 using the properties of modular arithmetic:
(n - 5) ≡ 0 (mod 21)
(n - 5) ≡ 0 (mod 7)
This means that n - 5 is congruent to 0 modulo 7.
Calculating the remainder
We can now solve for n by finding a number that satisfies the congruence:
n - 5 ≡ 0 (mod 7)
Adding 5 to both sides:
n ≡ 5 (mod 7)
This tells us that the remainder when n is divided by 7 is 5.
Conclusion
The remainder when the number n is divided by 7 is 5.