Question

A solution in a book demonstrates the following:

79z = 92x + 9y (values for z, x and y are in between 0 and 10)


Modulo 9, this gives -2z ≡ 2x and by multiplying both sides by 5 gives -z ≡ x. Since 1 ≤ x ≤ 9, we have z = 9 - x.



Can someone please explain how taking Modulo 9 of the equation above leads to these results in detail. Thanks.

Answer 1

In modular arithmetic, the modulo operation is used to find the remainder when one number is divided by another. For example, the expression "9 % 3" would evaluate to 0 because 9 divided by 3 leaves a remainder of 0.

In the given equation, taking the modulo 9 of both sides gives:

79z % 9 ≡ 92x + 9y % 9

The modulo operation distributes over addition, so we can simplify this to:

(79z % 9) ≡ (92x % 9) + (9y % 9)

Since any number divided by 9 has a remainder of itself, we can simplify this further to:

(z % 9) ≡ (2x % 9) + (y % 9)

This gives us the result:

-2z % 9 ≡ 2x % 9

Multiplying both sides by 5 gives:

(-2z % 9) * 5 ≡ (2x % 9) * 5

Which simplifies to:

-z % 9 ≡ x % 9

Since x is between 0 and 10, we know that x % 9 is between 0 and 9. Therefore, we can conclude that z = 9 - x.