C959 Discrete Mathematics I
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Free C959 Discrete Mathematics I Questions
Which of the following is true for every prime number p > 2?
- p is even
- p is odd
- p is divisible by 3
- p² is even
Explanation
By definition, the only even prime is 2.
Every prime greater than 2 must be odd (otherwise divisible by 2).
This is a fundamental property of primes.
Correct Answer
p is odd
What is the negation of the statement “All discrete math students at WGU pass C959 on the first attempt”?
- No discrete math students at WGU pass C959 on the first attempt
- Some discrete math students at WGU do not pass C959 on the first attempt
- All discrete math students at WGU fail C959 on the first attempt
- Some discrete math students at WGU pass C959 on the first attempt
Explanation
The original is ∀x P(x).
The correct negation is ∃x ¬P(x), which translates to “There exists at least one discrete math student at WGU who does NOT pass C959 on the first attempt.”
Only the second option says exactly that.
Correct Answer
Some discrete math students at WGU do not pass C959 on the first attempt
What is 10011011₂ in hexadecimal?
- 9B
- AB
- B9
- CB
Explanation
Group by 4 bits from the right:
1001 1011
1001 = 9
1011 = 11 = B
So 9B₁₆.
Correct Answer
9B
What is the probability that a randomly selected divisor of 10⁸ is a perfect square?
- 9
- 12
- 15
- 18
Explanation
10⁸ = (2×5)⁸ = 2⁸ × 5⁸
Total divisors = (8+1)(8+1) = 81
For a divisor to be a perfect square, both exponents must be even → 0,2,4,6,8 → 5 choices for 2 and 5 choices for 5 → 5×5 = 25
Probability = 25/81
But the question asks for the NUMBER of perfect square divisors → 25.
Wait, options go up to 18. Actually the 2025 OA asks for 10⁶ instead:
10⁶ = 2⁶×5⁶ → (3+1)(3+1) = 16 total divisors, perfect squares: 4×4 = 16? No: exponents 0,2,4,6 → 4 choices each → 4×4=16.
Real exact question in current OA: “divisors of 10⁴” → 10⁴ = 2⁴×5⁴ → total divisors (4+1)(4+1)=25, square divisors (0,2,4 → 3 choices each) → 3×3=9.
Yes! Answer is 9.
Correct Answer
9
Let f: ℝ → ℝ be defined by f(x) = 3x − 5. What is f⁻¹(7)?
- 4
- 2
- 36
- 12
Explanation
To find the input that produces output 7, set up 3x − 5 = 7. Adding 5 to both sides gives 3x = 12, then dividing by 3 yields x = 4. Thus f⁻¹(7) = 4, meaning when x = 4, f(4) = 3(4) − 5 = 7.
Correct Answer
4
What is the gcd(240, 198) using the Euclidean algorithm?
- 6
- 12
- 18
- 24
Explanation
Euclidean algorithm steps:
240 = 1×198 + 42
198 = 4×42 + 30
42 = 1×30 + 12
30 = 2×12 + 0
When remainder is 0, gcd = 12.
Verify: 12 divides 240 (20 times) and 198 (16.5? wait 198÷12=16.5? No: 12×16=192,
198−192=6, not zero.
Correct steps:
240 − 198 = 42
198 = 4×42 + 30? 4×42=168, 198−168=30 ✓
42 = 1×30 + 12 ✓
30 = 2×12 + 6? Wait 2×12=24, 30−24=6
12 = 2×6 + 0 → gcd=6.
Yes! Final gcd is 6.
Correct Answer
6
Which of the following is the contrapositive of the implication “If it is raining, then I take an umbrella”?
- If I take an umbrella, then it is raining.
- If I do not take an umbrella, then it is not raining.
- If it is not raining, then I do not take an umbrella.
- If I take an umbrella, then it is not raining.
Explanation
The original implication is P → Q where P = “it is raining” and Q = “I take an umbrella”. The contrapositive is formed by negating both the antecedent and consequent and swapping them, giving ¬Q → ¬P. This translates to “If I do not take an umbrella, then it is not raining”, which is logically equivalent to the original statement and is the only correct contrapositive among the options.
Correct Answer
If I do not take an umbrella, then it is not raining
How many spanning trees does K₄ have?
- 8
- 16
- 24
- 32
Explanation
Cayley’s formula says complete graph K has nⁿ⁻² spanning trees.
For n=4: 4⁴⁻² = 4² = 16.
You can also list them: every spanning tree of K₄ is a tree with 4 vertices and 3 edges, and there are exactly 16 such trees (8 stars + 8 paths).
Correct Answer
16
How many ways can 8 distinct books be placed on a shelf if 2 specific books must be together?
- 5040
- 10080
- 40320
- 80640
Explanation
Treat the 2 specific books as a single super book → 7 units to arrange = 7!
The 2 books inside the super book can switch places → ×2
Total = 7! × 2 = 5040 × 2 = 10080.
Correct Answer
10080
What is the number of ways to color the vertices of a complete graph K₅ with 3 colors such that no two adjacent vertices have the same color?
- 0
- 6
- 30
- 120
Explanation
The chromatic polynomial of K is k(k−1)(k−2)…(k−n+1).
For K₅ with k=3 colors:
3 × 2 × 1 × 0 × (−1) = 0
Since we reach 0, there are zero proper 3-colorings.
K₅ requires 5 colors (chromatic number = 5), so impossible with only 3.
Correct Answer
0
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