when z is divided by 8, the remainder is 5. What is the remainder when 4z is divided by 8. Can you please explain this to me step by step? I'm having difficulty understanding why the answer is 4

z/8=n+ 5/8

multiply by 4
4z/8=4n+20/8
but now you have a remainder of 20, and dividing it by 8 is 2r4
4z/8=(4n+2)+4/8
The term at the end is the remainder.

Step 1: We are given that when z is divided by 8, the remainder is 5. Mathematically, this can be represented as z ≡ 5 (mod 8), where the symbol "≡" denotes "is congruent to" and "(mod 8)" indicates that we are considering the remainder when divided by 8.

Step 2: Now, we need to find the remainder when 4z is divided by 8. Mathematically, this can be expressed as 4z ≡ ? (mod 8).

Step 3: To find the remainder, we can multiply both sides of the congruence equation (z ≡ 5 (mod 8)) by 4 (the number multiplying z in 4z). This gives us 4z ≡ 4 * 5 ≡ 20 (mod 8).

Step 4: Now, we need to simplify 20 (mod 8) to find its equivalent remainder when divided by 8.

Step 5: Divide 20 by 8: 20 ÷ 8 = 2 with a remainder of 4.

Step 6: Therefore, the remainder when 4z is divided by 8 is 4.

To find the remainder when 4z is divided by 8, we need to apply the properties of remainders and use some algebraic manipulation. Here's the step-by-step explanation:

1. We are given that when z is divided by 8, the remainder is 5. This can be written as z ≡ 5 (mod 8), where "≡" denotes congruence.
2. Now, let's multiply both sides of the congruence by 4. This yields 4z ≡ 20 (mod 8). This step is valid because multiplying both sides of a congruence by the same number does not change the congruence relationship.
3. Simplifying the right side of the congruence, we have 20 ≡ 4 (mod 8). This means that 20 and 4 are congruent to each other modulo 8.
4. The definition of congruence modulo is that two numbers are congruent modulo n if their difference is divisible by n. In this case, 20 - 4 is equal to 16, which is divisible by 8 (16 ÷ 8 = 2).
5. Therefore, we can conclude that 20 and 4 are indeed congruent to each other modulo 8. In other words, 20 ≡ 4 (mod 8).
6. Now, going back to the original question, we can substitute 4z ≡ 20 (mod 8) with the congruent expression we just derived: 4z ≡ 4 (mod 8).
7. This implies that when 4z is divided by 8, the remainder is 4, since 4 is the value on the right side of the congruence.

Thus, the remainder when 4z is divided by 8 is 4.