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Questions and Answers
What does the notation $a \mid b$ signify in the context of divisibility?
What does the notation $a \mid b$ signify in the context of divisibility?
- a is divisible by b
- a is not divisible by b
- b is divisible by a (correct)
- a and b are equal
If $a \mid b$, then $a \mid -b$ is always true.
If $a \mid b$, then $a \mid -b$ is always true.
True (A)
If $a \mid b$ and $b \mid a$ for integers a and b, what is the relationship between a and b?
If $a \mid b$ and $b \mid a$ for integers a and b, what is the relationship between a and b?
a = ±b
In the expression b = aq + r, where a, b, q, and r are integers, r represents the __________ when b is divided by a.
In the expression b = aq + r, where a, b, q, and r are integers, r represents the __________ when b is divided by a.
Match the divisibility statements with their corresponding interpretations:
Match the divisibility statements with their corresponding interpretations:
Which of the following statements is equivalent to $a \mid b$?
Which of the following statements is equivalent to $a \mid b$?
If a number 'a' is divisible by another number 'b', then 'a' is a multiple of 'b'.
If a number 'a' is divisible by another number 'b', then 'a' is a multiple of 'b'.
According to the definition provided, if a and b are integers and $a \neq 0$, what must exist for 'a divides b' to be true?
According to the definition provided, if a and b are integers and $a \neq 0$, what must exist for 'a divides b' to be true?
If 2|6, then there exists an integer such that 6 = 2 * _____.
If 2|6, then there exists an integer such that 6 = 2 * _____.
If $a \mid b$ and $a \mid c$, which of the following is also true for any integers x and y?
If $a \mid b$ and $a \mid c$, which of the following is also true for any integers x and y?
If $a \mid b$ and $c \mid d$, then $ac \mid (b + d)$ is always true.
If $a \mid b$ and $c \mid d$, then $ac \mid (b + d)$ is always true.
What are the only divisors of 1?
What are the only divisors of 1?
If $a \mid b$, then the absolute value of a is ________ than or equal to the absolute value of b.
If $a \mid b$, then the absolute value of a is ________ than or equal to the absolute value of b.
Which of the following is the correct reading of the notation 2|6?
Which of the following is the correct reading of the notation 2|6?
If 5 does not divide 17, we can write 5 ł 17.
If 5 does not divide 17, we can write 5 ł 17.
If $a \mid b$ and $b \mid c$, what property of divisibility allows us to conclude that $a \mid c$?
If $a \mid b$ and $b \mid c$, what property of divisibility allows us to conclude that $a \mid c$?
The statement '21 is divisible by 3' can be written in notation as 3 ______ 21.
The statement '21 is divisible by 3' can be written in notation as 3 ______ 21.
Given that 'a' divides 'b', which expression correctly represents 'b' in terms of 'a' and some integer 'c'?
Given that 'a' divides 'b', which expression correctly represents 'b' in terms of 'a' and some integer 'c'?
The expression 8 ł 4 indicates that 8 is a factor of 4.
The expression 8 ł 4 indicates that 8 is a factor of 4.
What does $\forall$ symbol stand for?
What does $\forall$ symbol stand for?
If 2|4, then, according to one of the illustrations, 2|4k for any integer ____.
If 2|4, then, according to one of the illustrations, 2|4k for any integer ____.
Which of the following notations represents that a does not divide b?
Which of the following notations represents that a does not divide b?
5 | 0
5 | 0
If we have $a \mid b$ and $c \mid d$, then what expression must $ac$ divide?
If we have $a \mid b$ and $c \mid d$, then what expression must $ac$ divide?
We can use mathematical _________ for the shortcuts of sentences.
We can use mathematical _________ for the shortcuts of sentences.
Flashcards
Definition of 'a divides b'
Definition of 'a divides b'
A number 'a' divides 'b' if there exists an integer 'c' such that b = ac.
Divisibility in Algebra
Divisibility in Algebra
In algebra, divisibility and multiples are frequently encountered.
What does 'a | b' mean?
What does 'a | b' mean?
Notation used to indicate that a number 'a' divides 'b'.
What does 'a ł b' mean?
What does 'a ł b' mean?
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Equivalent statements
Equivalent statements
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Theorem: If a|b, then a|bc
Theorem: If a|b, then a|bc
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Corollary: If a|b, then a|-b
Corollary: If a|b, then a|-b
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Transitive Property of Divisibility
Transitive Property of Divisibility
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Theorem: If a|b and c|d, then ac|bd
Theorem: If a|b and c|d, then ac|bd
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Theorem: If a|b, then |a| ≤ |b|
Theorem: If a|b, then |a| ≤ |b|
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Divisors of 1
Divisors of 1
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Corollary: If a|b and b|a, then a = ±b
Corollary: If a|b and b|a, then a = ±b
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If a|b and a|c, then...
If a|b and a|c, then...
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Divisibility check
Divisibility check
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Study Notes
- Math relies on symbols and notations for its language.
- Understanding these symbols is crucial because a slight change can alter the meaning.
- Notations can serve as shortcuts for sentences, allowing interpretations through symbols.
- The text is a presentation by Ms. Angelyn T. Natividad of Tanauan Institute.
Divisibility
- For any integer a, it can be written as a ∈ Z.
- For any pair of integers a and b, it can be written as ∀ a, b ∈ Z.
- In algebra, encounter terms like divisibility or multiple.
- The phrases are: 2|6 or 6 is divisible by 2; the notation is read as "2 divides 6".
- If a number b is not divisible by another number a, it is written as a ł b.
- If a and b are in Z and a ≠ 0, then a divides b, denoted as a|b.
- This means there's an integer c such that b = ac.
Examples of Divisibility
- 3|12 because c = 4, and 3c = 3(4) = 12.
- Equivalent statements for a|b:
- b is divisible by a
- a divides b
- b is a multiple of a
- a is a divisor/factor of b
- If a does not divide b (a ł b), integers q and r exist such that b = aq + r and 0 < r < a.
- 2|6, since 6 = 2(3).
- -32|64, since 64 = 32(-2).
- 25|-125, since -125 = 25(-5).
- -21|-84, since -84 = -21(4).
- 97|0, since 0 = 97(0).
- 4|4, 2|6, 6|6 are all examples of divisibility
- 8 ł 4, 10 ł 5, 14 ł 7 are examples where the first number does not divide the second.
- Questions to check divisibility:
- Is 21 divisible by 3?
- Does 5 divide 40?
- Does 7|42?
- Is 32 a multiple of -16?
- Is 6 a factor of 54?
- Is 7 a factor of -7?
Theorem of Divisibility
- For any integers a, b, and c: if a|b, then a|bc.
- Proof: If a|b, then aq = b for some integer q ∈ Z. Therefore, a(cq) = bc, and since cq ∈ Z, then a|bc.
- Illustration: if 2|4, then 2|4k for any integer k.
- For a, b ∈ Z, if a|b, then a|-b.
- Illustration: if 2|4, then 2|-4.
- For a|b and b|c, where a, b, and c are integers, then a|c. This is known as the transitive property of divisibility.
- Given a|b and b|c, integers q1 and q2 exist such that b = aq1 and c = bq2. Substituting, c = a(q1q2), and since q1q2 ∈ Z, then a|c.
- If a|b and c|d for any integers a, b, c, and d, then ac|bd.
- If a|b and c|d, integers q1 and q2 exist where b = aq1 and d = cq2. By substitution, bd = aq1 * cq2, thus bd = ac(q1q2), and since q1q2 ∈ Z, then ac|bd.
- Illustration: if a=2, b=6, c=5 and d=10; as 2|6 and 5|10 then ac|bd, it becomes 10|60
Inequalities
- For a|b for any integers a and b, then |a| ≤ |b|.
- Proof: if a|b, then b = aq for some integer q; therefore, |b| = |aq| = |a||q|.
- Given b ≠ 0, then q ≠ 0, hence |q| ≥ 1. Multiplying by |a| yields |b| = |a||q| ≥ |a|.
Additional Notes
- Illustration: 2-4 then 2<|-4|; -2|-4 then|2| ≤ |4|
- The the only divisors of 1 are 1 and -1
- For any integers a and b: if a|b and b|a, then a = ±b.
- Illustration: 2|-2 and -2|2, then 2 = |-2|.
- For any integers a, b, and c: if a|b and a|c, then a|bx + cy for any integers x and y.
- Proof: ifa|b and a|c, b = aq₁ and c = aq₂. So, bx + cy = aq₁x + aq₂y = a(q₁x + q₂y).
- Illustration: if 2|4 and 2|6, then 2|4k₁ + 6k₂ for any integers k₁ and k₂.
Example of Algebraic Expression
- If a and b are integers, is 3a+3b divisible by 3?
- If k and m are integers, is 10km divisible by 5?
- Does 4|15 for checking nondivisibility?
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