1. when z is divided by 8, the remainder is 5. What is the remainder when 4z is divided by 8 ?

2. If n is an integer, which of the following must be odd ?

A. 3n-5
B. 3n + 4
C. 4n + 10
D. 4n - 5
E. 5n + 7

1. To find the answer, we need to understand how remainders work when numbers are divided.

Let's work through it step-by-step:

a. When z is divided by 8, the remainder is 5. This can be written as:
z = 8a + 5, where 'a' is any integer.

b. Now, we want to find the remainder when 4z is divided by 8. We can substitute the value of z from the previous equation into this one:
4z = 4(8a + 5).

c. Expanding the equation, we get:
4(8a + 5) = 32a + 20.

d. To find the remainder when 4z is divided by 8, we divide 32a + 20 by 8:
(32a + 20) ÷ 8 = 4a + 2 remainder 4.

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

2. To determine which expressions must be odd when n is an integer, we need to understand the properties of odd and even numbers.

a. An odd number is any number that cannot be divided evenly by 2. It always leaves a remainder of 1 when divided by 2.
b. An even number is any number that can be divided evenly by 2. It leaves a remainder of 0 when divided by 2.

Let's analyze each expression to see which ones will always be odd, regardless of the value of n:

A. 3n - 5:
Here, we have a constant term (-5) subtracted from a multiple of 3 (3n). Subtraction does not affect the odd/even properties, so we only need to consider 3n. Since every multiple of 3 is odd, regardless of the value of n, the expression 3n - 5 will also be odd.

B. 3n + 4:
Similar to the previous case, we only need to consider 3n. As every multiple of 3 is odd, regardless of the value of n, the expression 3n + 4 will always be even.

C. 4n + 10:
This expression contains a constant term (10) added to a multiple of 4 (4n). Addition does not affect the odd/even properties, so we only need to consider 4n. Every multiple of 4 is even regardless of the value of n, so the expression 4n + 10 will always be even.

D. 4n - 5:
Similar to case C, we only need to consider 4n. Every multiple of 4 is even regardless of the value of n, so the expression 4n - 5 will always be odd.

E. 5n + 7:
Here, we only need to consider 5n. Every multiple of 5 is odd regardless of the value of n. Therefore, the expression 5n + 7 will always be odd.

To summarize, the expressions that must be odd when n is an integer are:
A. 3n - 5
D. 4n - 5
E. 5n + 7