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Questions and Answers
Which of the following statements about the greatest common divisor (G.C.D) is true?
Which of the following statements about the greatest common divisor (G.C.D) is true?
- The G.C.D of any two integers can be zero.
- The G.C.D is not always expressible as a linear combination of the two integers.
- The G.C.D of any two integers is always positive. (correct)
- The G.C.D can only be defined for positive integers.
If P is a prime number and P divides the product of two integers a and b, it necessarily follows that P divides at least one of those integers.
If P is a prime number and P divides the product of two integers a and b, it necessarily follows that P divides at least one of those integers.
True (A)
What does it mean for a function f: X→Y to be surjective?
What does it mean for a function f: X→Y to be surjective?
A function f is surjective if every element in the codomain Y is mapped to by at least one element in the domain X.
If A and B are reflexive relations, then their union is also ______.
If A and B are reflexive relations, then their union is also ______.
Match the following functions with their properties:
Match the following functions with their properties:
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Study Notes
Exam Information
- Subject: Algebra I (USMT102)
- Date: November 10, 2023
- Exam Duration: 2 hours, from 10:00 to 12:30 PM
- Maximum Marks: 75
- All questions are mandatory
Question Breakdown
Q1: Number Theory
- G.C.D of Integers: Any two non-zero integers a and b have a greatest common divisor (g.c.d), expressible as ( ma + nb ) for integers m and n.
- Example Problems:
- Calculate G.C.D of 2210 and 357, express as ( 2210x + 357y ).
- Prove ( 281 \equiv 2 \mod 41 ) using Fermat's Little Theorem.
- If P is a prime and divides the product ab, it also divides a or b.
Q2: Functions and Relations
- Function Properties: If ( g \circ f ) is bijective and f is surjective, function g is injective.
- Example Problems:
- Prove f: R→R defined by ( f(x) = 2x + 5 ) is a bijective function.
- Define binary operation ( a * b = a + 2b ) on Z, check for commutativity and associativity.
- Analyze the relation R among students in college for reflexivity, symmetry, and transitivity.
Q3: Polynomials
- Polynomial Definition: Polynomial ( f(x) = a_n x^n + a_{n-1}x^{n-1} +...+ a_1x + a_0 ) has integer coefficients, where each ( a_i \in Z ).
- Rational Roots Theorem: If a rational number ( \frac{p}{q} ) is a root of f(x), then ( \frac{p}{a_n} ) and ( \frac{q}{a_0} ) must also hold.
- Example Problems:
- Find quotient and remainder of ( x^3 - 3x^2 + 4x + 8 ) when divided by ( x^2 + 2 ).
- Determine the multiplicity of each root for ( f(x) = x^3 - 4x^2 + 5x - 2 ).
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