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Difficulty: HardOdd and Even Integers (Parity)

For how many integers nn satisfying 0≤n≤1000 \le n \le 100 is the expression n4+2n+nn^4 + 2^n + n an odd integer?

Answer: 1

Answer

There is exactly 1 value of nn (specifically n=0n = 0) in the range 0≤n≤1000 \le n \le 100 for which the expression n4+2n+nn^4 + 2^n + n is an odd integer.
For any integer n≥1n \ge 1, n4n^4 and nn share the exact same parity (both even or both odd), making n4+nn^4 + n always even. Additionally, for n≥1n \ge 1, 2n2^n is an even integer. The sum of two even integers is always even, so n4+2n+nn^4 + 2^n + n is even for all 1≤n≤1001 \le n \le 100. When n=0n = 0, however, 20=12^0 = 1 is odd and 04+0=00^4 + 0 = 0 is even. The sum 0+1=10 + 1 = 1 is odd. Thus, n=0n = 0 is the single value that satisfies the condition.

Step-by-Step Solution

1
Analyze parity of n4+nn^4 + n
n4+nn^4 + n is even for all integers nn
An integer nn and its power n4n^4 always have the same parity. The sum of two integers of the same parity is always even.
2
Evaluate 2n2^n parity for n≥1n \ge 1
2n2^n is even for all n≥1n \ge 1
Any positive integer power of 2 contains 2 as a prime factor and is therefore even.
3
Evaluate the expression for n=0n = 0
04+20+0=10^4 + 2^0 + 0 = 1, which is odd
Zero is a non-negative integer within 0≤n≤1000 \le n \le 100. By definition of exponents, 20=12^0 = 1, which is odd.
4
Combine results and count valid integers
Total count = 1
For n≥1n \ge 1, even + even = even. For n=0n = 0, even + odd = odd. Hence, only n=0n = 0 produces an odd integer.

Key Concept

Parity Rules for Exponent Base 2 Edge Cases and Algebraic Expressions
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Topics

Odd and Even Integers (Parity)

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