Examkin

Question

Difficulty: EasyOdd and Even Integers (Parity)

If pp and qq are integers such that 3p+2q3p + 2q is an odd integer, which of the following statements must be true?

  1. pp must be odd, but qq can be any integerAnswer
  2. B
    pp must be odd, and qq must be even
  3. C
    Both pp and qq must be odd
  4. D
    pp must be even, and qq must be odd
  5. E
    Both pp and qq must be even

Answer

pp must be odd, but qq can be any integer
Multiplying any integer qq by 2 guarantees that 2q2q is even. For the sum 3p+2q3p + 2q to be odd, 3p3p must be odd because the sum of an odd integer and an even integer is odd. Since 3 is odd, 3p3p is odd if and only if pp is odd. The variable qq can be any integer because 2q2q remains even whether qq is even or odd.

Step-by-Step Solution

1
Analyze the parity of the term 2q2q
For any integer qq, 2q2q is a multiple of 2, so 2q2q is always an even integer.
The product of an even integer and any integer is always even.
2
Determine the required parity of the term 3p3p
Since 3p+2q3p + 2q is odd and 2q2q is even, 3p3p must be odd.
An odd integer plus an even integer yields an odd sum (odd+even=odd\text{odd} + \text{even} = \text{odd}).
3
Determine the parity of pp
Since 3p3p is odd and 3 is odd, pp must be an odd integer.
The product of two integers is odd if and only if both integers are odd (odd×odd=odd\text{odd} \times \text{odd} = \text{odd}).
4
Evaluate the restrictions on qq
qq can be any integer (even or odd).
Multiplying qq by 2 forces 2q2q to be even regardless of whether qq is odd or even.

Key Concept

Parity rules under addition and multiplication of integers
Rate this question

Topics

Odd and Even Integers (Parity)

More questions from this topic