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Number Theory Module 2: Divisibility and Diophantine Equations

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Number Theory Module 2: Divisibility and Diophantine Equations

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Questions and Answers

Which equation has a solution where $X = 170$ and $Y = 357?

$540X + 630Y = 270$

$170X + 357 = 34$ (correct)

$6630X + 7956Y = 1326$

$23716X + 75460Y = 6468$

If $b$ divides the difference $r - s$, where $0 \leq r \leq b$ and $0 \leq s \leq b$, then what can be concluded about $r$ and $s?

$r$ and $s$ can be either equal or different

$r$ and $s$ must be equal (correct)

$r$ and $s$ must be different

No conclusion can be drawn about the relationship between $r$ and $s

Let $a$, $b$, and $c$ be integers with $c \neq 0$. Which of the following statements is true?

$c$ divides $a - b$ if and only if $a$ and $b$ have different remainders after division by $c$

$c$ divides $a - b$ if and only if $a$ and $b$ have the same remainder after division by $c$ (correct)

$c$ divides $a - b$ if and only if $a$ and $b$ are both multiples of $c$

$c$ divides $a - b$ if and only if $a$ and $b$ are both divisible by $c$

Let $a$, $b$, and $c$ be integers with $a$, $b$, $c \neq 0$ and $c \neq 0$. Which of the following is true about the greatest common divisor (GCD) of $ca$ and $cb$?

The GCD of $ca$ and $cb$ is $c$ times the GCD of $a$ and $b$ Signup and view all the answers

Which of the following equations is an example of a Diophantine equation?

All of the above are examples of Diophantine equations Signup and view all the answers

What is the relationship between the remainders of $a$ and $b$ after division by $c$, if $c$ divides $a - b$?

The remainders must be the same Signup and view all the answers

Which of the following statements about the greatest common divisor (GCD) of $ca$ and $cb$, where $a$, $b$, and $c$ are integers with $c \neq 0$, is false?

The GCD of $ca$ and $cb$ is always $1 Signup and view all the answers

Number Theory Module 2: Divisibility and Diophantine Equations

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Questions and Answers

Which equation has a solution where $X = 170$ and $Y = 357?

If $b$ divides the difference $r - s$, where $0 \leq r \leq b$ and $0 \leq s \leq b$, then what can be concluded about $r$ and $s?

Let $a$, $b$, and $c$ be integers with $c \neq 0$. Which of the following statements is true?

Let $a$, $b$, and $c$ be integers with $a$, $b$, $c \neq 0$ and $c \neq 0$. Which of the following is true about the greatest common divisor (GCD) of $ca$ and $cb$?

Which of the following equations is an example of a Diophantine equation?

What is the relationship between the remainders of $a$ and $b$ after division by $c$, if $c$ divides $a - b$?

Which of the following statements about the greatest common divisor (GCD) of $ca$ and $cb$, where $a$, $b$, and $c$ are integers with $c \neq 0$, is false?

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