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
To find the remainder when dividing by a number, we can use the modulo operator (%). The modulo operator gives us the remainder when one number is divided by another.
1. We are given that when z is divided by 8, the remainder is 5. In mathematical notation, we can write this as z % 8 = 5.
To find the remainder when 4z is divided by 8, we can substitute the value of z from the given information:
4z % 8 = 4 * (z % 8) = 4 * 5 = 20
So, when 4z is divided by 8, the remainder is 20.
2. We need to determine which expression among A, B, C, D, and E must be odd when n is an integer.
For an integer to be considered odd, it must not be divisible by 2. Therefore, an expression is odd if the coefficient of n (the number multiplying n) is odd.
Let's analyze each option:
A. 3n - 5: The coefficient of n is 3, which is odd. Therefore, this expression is always odd.
B. 3n + 4: The coefficient of n is 3, which is odd. Therefore, this expression is always odd.
C. 4n + 10: The coefficient of n is 4, which is even. Therefore, this expression is not always odd.
D. 4n - 5: The coefficient of n is 4, which is even. Therefore, this expression is not always odd.
E. 5n + 7: The coefficient of n is 5, which is odd. Therefore, this expression is always odd.
From the analysis, options A and E must be odd expressions when n is an integer.