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.