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Zorluk: Çok zorOdd and Even Integers (Parity)

If mm, nn, and pp are integers such that m3n−n2pm^3 n - n^2 p is an odd integer and m(n+p)m(n + p) is an even integer, which of the following expressions MUST be an even integer?

  1. m2+n+pm^2 + n + pCevap
  2. B
    m+npm + np
  3. C
    mn+pmn + p
  4. D
    mp+nmp + n
  5. E
    n+p+1n + p + 1

Cevap

m2+n+pm^2 + n + p must be an even integer.
Factoring m3n−n2pm^3 n - n^2 p as n(m3−np)n(m^3 - np) shows that nn is odd and m3−npm^3 - np is odd. Because nn is odd, m3−npm^3 - np has the same parity as m−pm - p, meaning mm and pp have opposite parities. Testing the condition that m(n+p)m(n + p) is even reveals that pp must be odd and mm must be even (if pp were even, mm would be odd and m(n+p)m(n + p) would be odd, a contradiction). With mm even, nn odd, and pp odd, the expression m2+n+pm^2 + n + p calculates as even+odd+odd=even\text{even} + \text{odd} + \text{odd} = \text{even}, which must be an even integer.

Adım Adım Çözüm

1
Analyze the parity of the expression m3n−n2pm^3 n - n^2 p.
nn is odd and m3−npm^3 - np is odd.
The expression can be factored as n(m3−np)n(m^3 - np). For a product of two integers to be odd, both factors must be odd. Hence, nn must be odd, and m3−npm^3 - np must also be odd.
2
Determine the relative parities of mm and pp.
mm and pp must have opposite parities (one is even, the other is odd).
Since nn is odd, npnp has the same parity as pp. The expression m3−npm^3 - np has the same parity as m−pm - p. For m−pm - p to be odd, mm and pp must have opposite parities.
3
Use the second given condition m(n+p)m(n + p) is even to determine the exact parity of mm and pp.
mm is even and pp is odd.
If pp were even, then n+p=odd+even=oddn + p = \text{odd} + \text{even} = \text{odd}. Since mm and pp have opposite parities, mm would be odd, making m(n+p)=odd×odd=oddm(n + p) = \text{odd} \times \text{odd} = \text{odd}, which contradicts the condition that m(n+p)m(n + p) is even. Thus, pp cannot be even; pp must be odd, which implies mm is even.
4
Evaluate the parity of the options using m=evenm = \text{even}, n=oddn = \text{odd}, and p=oddp = \text{odd}.
m2+n+p=even+odd+odd=evenm^2 + n + p = \text{even} + \text{odd} + \text{odd} = \text{even}.
m2m^2 is even since mm is even. Summing an even number (m2m^2) and two odd numbers (nn and pp) yields an even integer.

Anahtar Kavram

Parity rules for integer addition and multiplication: the product of integers is odd if and only if all factors are odd, and the sum of two odd integers is even.
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Konular

Odd and Even Integers (Parity)

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